UBC Wiki - User contributions [en]

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T02:04:25Z

StephanieUrness:

'''<big>Richter scale and it's relation to logarithmic functions:</big>'''

Simply put the Richter scale measures the amount of energy released by an earthquake, and translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations(waves) in the earth, by an instrument known as the "seismograph". Measuring the highest amplitude of the vibrations/waves we can calculate how much energy was released, it goes to say the more energy released, the higher the amplitude of the waves. In essence the amplitude measured will tell us the amount of energy released, which in turn will tell us the magnitude/strength of the earthquake.

But how do we go from amplitude to energy to magnitude?

Magnitude, when simplified as numbers from 1 - 10, does not give the actual calculated magnitude of an earthquake. Each increment increases by a factor of one, when in reality each increase (4 to 5, or 5 to 6) indicates an intensity 10 times stronger than the previous number.
Ie: Earthquake of magnitude 5 is 10 times stronger than one of magnitude 4.
These numbers increase exponentially, and not being so user friendly, it is scaled from 1-10.

Formula given for the Richter scale is:

<math> R = Log\frac{I}{I_o} </math>

Using your logarithm rules it can also be written as:

<math> R = Log_{10}I-Log_{10}I_o </math>

<math> I = </math> amplitude of the earthquake (taken from 100km from where the vibrations originated - measured from the surface above where the vibrations originated).

<math> I_o = </math> Intensity of the smallest sized activity measurable, hardly detectable (if you were to actually have an earthquake this size and put it on the Richter scale it would = 0. Can you figure out why by looking at the formula?). This is denoted as 1 micron or <math>10^{-4}</math> cm

Why do we need <math> I_o = </math> . It gives us a reference value relative to our actual measured earthquake.

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

References:

http://www.sosmath.com/algebra/logs/log5/log56/log56.html

http://en.wikipedia.org/wiki/Richter_magnitude_scale

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T02:00:30Z

StephanieUrness:

'''<big>Richter scale and it's relation to logarithmic functions:</big>'''

Simply put the Richter scale measures the amount of energy released by an earthquake, and translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations(waves) in the earth, by an instrument known as the "seismograph". Measuring the highest amplitude of the vibrations/waves we can calculate how much energy was released, it goes to say the more energy released, the higher the amplitude of the waves. In essence the amplitude measured will tell us the amount of energy released, which in turn will tell us the magnitude/strength of the earthquake.

But how do we go from amplitude to energy to magnitude?

Magnitude, when simplified as numbers from 1 - 10, does not give the actual calculated magnitude of an earthquake. Each increment increases by a factor of one, when in reality each increase (4 to 5, or 5 to 6) indicates an intensity 10 times stronger than the previous number.
Ie: Earthquake of magnitude 5 is 10 times stronger than one of magnitude 4.
These numbers increase exponentially, and not being so user friendly, it is scaled from 1-10.

Formula given for the Richter scale is:

<math> R = Log\frac{I}{I_o} </math>

Using your logarithm rules it can also be written as:

<math> R = Log_{10}I-Log_{10}I_o </math>

<math> I = </math> amplitude of the earthquake (taken from 100km from where the vibrations originated - measured from the surface above where the vibrations originated).

<math> I_o = </math> Intensity of the smallest sized activity measurable, hardly detectable (if you were to actually have an earthquake this size and put it on the Richter scale it would = 0. Can you figure out why by looking at the formula?). This is denoted as 1 micron or <math>10^{-4}</math> cm

The reason for

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

References:

http://www.sosmath.com/algebra/logs/log5/log56/log56.html

http://en.wikipedia.org/wiki/Richter_magnitude_scale

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T01:37:03Z

StephanieUrness:

'''<big>Richter scale and it's relation to logarithmic functions:</big>'''

Simply put the Richter scale measures the amount of energy released by an earthquake, and translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations(waves) in the earth, by an instrument known as the "seismograph". Measuring the highest amplitude of the vibrations/waves we can calculate how much energy was released, it goes to say the more energy released, the higher the amplitude of the waves. In essence the amplitude measured will tell us the amount of energy released, which in turn will tell us the magnitude/strength of the earthquake.

But how do we go from amplitude to energy to magnitude?

Magnitude, when simplified as numbers from 1 - 10, does not give the actual calculated magnitude of an earthquake. Each increment increases by a factor of one, when in reality each increase (4 to 5, or 5 to 6) indicates an intensity 10 times stronger than the previous number.
Ie: Earthquake of magnitude 5 is 10 times stronger than one of magnitude 4.
These numbers increase exponentially, and not being so user friendly, it is scaled from 1-10.

Formula given for the Richter scale is:

<math> R = Log\frac{I}{I_o} </math>

Using your logarithm rules can also be written as:

<math> R = Log_{10}I-Log_{10}I_o </math>

<math> I = </math> amplitude of the earthquake (taken from 100km from where the vibrations originated - measured from the surface above where the vibrations originated).

<math> I_o = </math> Intensity of a very small earthquake, movement is hardly detectable (if you were to actually have an earthquake this size and put it on the Richter scale it would = 0. Can you figure out why by looking at the formula?). This is denoted as 1 micron or <math>10^{-4}</math> cm

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

References:

http://www.sosmath.com/algebra/logs/log5/log56/log56.html

http://en.wikipedia.org/wiki/Richter_magnitude_scale

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T01:36:09Z

StephanieUrness:

'''<big>Richter scale and it's relation to logarithmic functions:</big>'''

Simply put the Richter scale measures the amount of energy released by an earthquake, and translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations(waves) in the earth, by an instrument known as the "seismograph". Measuring the highest amplitude of the vibrations/waves we can calculate how much energy was released, it goes to say the more energy released, the higher the amplitude of the waves. In essence the amplitude measured will tell us the amount of energy released, which in turn will tell us the magnitude/strength of the earthquake.

But how do we go from amplitude to energy to magnitude?

Magnitude, when simplified as numbers from 1 - 10, does not give the actual calculated magnitude of an earthquake. Each increment increases by a factor of one, when in reality each increase (4 to 5, or 5 to 6) indicates an intensity 10 times stronger than the previous number.
Ie: Earthquake of magnitude 5 is 10 times stronger than one of magnitude 4.
These numbers increase exponentially, and not being so user friendly, it is scaled from 1-10.

Formula given for the Richter scale is:

<math> R = Log\frac{I}{I_o} </math>

which, using your logarithm rules can be written also like (with natural log base of 10:

<math> R = Log_{10}I-Log_{10}I_o </math>

<math> I = </math> amplitude of the earthquake (taken from 100km from where the vibrations originated - measured from the surface above where the vibrations originated).

<math> I_o = </math> Intensity of a very small earthquake, movement is hardly detectable (if you were to actually have an earthquake this size and put it on the Richter scale it would = 0. Can you figure out why by looking at the formula?). This is denoted as 1 micron or <math>10^{-4}</math> cm

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

References:

http://www.sosmath.com/algebra/logs/log5/log56/log56.html

http://en.wikipedia.org/wiki/Richter_magnitude_scale

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T01:35:39Z

StephanieUrness:

'''Richter scale and it's relation to logarithmic functions:'''

Simply put the Richter scale measures the amount of energy released by an earthquake, and translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations(waves) in the earth, by an instrument known as the "seismograph". Measuring the highest amplitude of the vibrations/waves we can calculate how much energy was released, it goes to say the more energy released, the higher the amplitude of the waves. In essence the amplitude measured will tell us the amount of energy released, which in turn will tell us the magnitude/strength of the earthquake.

But how do we go from amplitude to energy to magnitude?

Magnitude, when simplified as numbers from 1 - 10, does not give the actual calculated magnitude of an earthquake. Each increment increases by a factor of one, when in reality each increase (4 to 5, or 5 to 6) indicates an intensity 10 times stronger than the previous number.
Ie: Earthquake of magnitude 5 is 10 times stronger than one of magnitude 4.
These numbers increase exponentially, and not being so user friendly, it is scaled from 1-10.

Formula given for the Richter scale is:

<math> R = Log\frac{I}{I_o} </math>

which, using your logarithm rules can be written also like (with natural log base of 10:

<math> R = Log_{10}I-Log_{10}I_o </math>

<math> I = </math> amplitude of the earthquake (taken from 100km from where the vibrations originated - measured from the surface above where the vibrations originated).

<math> I_o = </math> Intensity of a very small earthquake, movement is hardly detectable (if you were to actually have an earthquake this size and put it on the Richter scale it would = 0. Can you figure out why by looking at the formula?). This is denoted as 1 micron or <math>10^{-4}</math> cm

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

References:

http://www.sosmath.com/algebra/logs/log5/log56/log56.html

http://en.wikipedia.org/wiki/Richter_magnitude_scale

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T01:32:44Z

StephanieUrness:

'''Richter scale and it's relation to logarithmic functions:'''

Simply put the Richter scale measures the amount of energy released by an earthquake, and translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations(waves) in the earth, by an instrument known as the "seismograph". Measuring the highest amplitude of the vibrations/waves we can calculate how much energy was released, it goes to say the more energy released, the higher the amplitude of the waves. In essence the amplitude measured will tell us the amount of energy released, which in turn will tell us the magnitude/strength of the earthquake.

But how do we go from amplitude to energy to magnitude?

Magnitude, when simplified as numbers from 1 - 10, does not give the actual calculated magnitude of an earthquake. Each increment increases by a factor of one, when in reality each increase (4 to 5, or 5 to 6) indicates an intensity 10 times stronger than the previous number.
Ie: Earthquake of magnitude 5 is 10 times stronger than one of magnitude 4.
These numbers increase exponentially, and not being so user friendly, it is scaled from 1-10.

Formula given for the Richter scale is:

<math> R = Log\frac{I}{I_o} </math>

which, using your logarithm rules can be written also like (with natural log base of 10:

<math> R = LogI-LogI_o </math>

<math> I = </math> amplitude of the earthquake (taken from 100km from where the vibrations originated - measured from the surface above where the vibrations originated).

<math> I_o = </math> Intensity of a very small earthquake, movement is hardly detectable (if you were to actually have an earthquake this size and put it on the Richter scale it would = 0. Can you figure out why by looking at the formula?). This is denoted as 1 micron or <math> 10^(-4) </math> cm

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

References:

http://www.sosmath.com/algebra/logs/log5/log56/log56.html

http://en.wikipedia.org/wiki/Richter_magnitude_scale

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T01:22:57Z

StephanieUrness:

'''Richter scale and it's relation to logarithmic functions:'''

Simply put the Richter scale measures the amount of energy released by an earthquake, and translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations(waves) in the earth, by an instrument known as the "seismograph". Measuring the highest amplitude of the vibrations/waves we can calculate how much energy was released, it goes to say the more energy released, the higher the amplitude of the waves. In essence the amplitude measured will tell us the amount of energy released, which in turn will tell us the magnitude/strength of the earthquake.

But how do we go from amplitude to energy to magnitude?

Magnitude, when simplified as numbers from 1 - 10, does not give the actual calculated magnitude of an earthquake. Each increment increases by a factor of one, when in reality each increase (4 to 5, or 5 to 6) indicates an intensity 10 times stronger than the previous number.
Ie: Earthquake of magnitude 5 is 10 times stronger than one of magnitude 4.
These numbers increase exponentially, and not being so user friendly, it is scaled from 1-10.

Formula given for the Richter scale is:

<math> R = Log\frac{I}{I_o} </math>

which, using your logarithm rules can be written also like (with natural log base of 10:

<math> R = LogA-LogA_o </math>

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T01:18:46Z

StephanieUrness:

'''Richter scale and it's relation to logarithmic functions:'''

Simply put the Richter scale measures the amount of energy released by an earthquake, and translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations(waves) in the earth, by an instrument known as the "seismograph". Measuring the highest amplitude of the vibrations/waves we can calculate how much energy was released, it goes to say the more energy released, the higher the amplitude of the waves. In essence the amplitude measured will tell us the amount of energy released, which in turn will tell us the magnitude/strength of the earthquake.

But how do we go from amplitude to energy to magnitude?

Magnitudes, although when simplified as numbers from 1 - 10, do not mean the actual magnitude of an earthquake increased by a factor of 1 each time. Each number indicates an intensity 10 times stronger than the previous number.
Ie: Earthquake of magnitude 5 is 10 times stronger than one of magnitude 4.

These numbers increasingly exponentially as you continue on, and are not so user friendly, one of the reasons it is scaled down from 1-10.

Formula given for the Richter scale is:

<math> R = Log\frac{I}{I_o} </math>

which, using your logarithm rules can be written also like (with natural log base of 10:

<math> R = LogA-LogA_o </math>

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T01:07:48Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T00:55:25Z

StephanieUrness:

'''Richter scale and it's relation to logarithmic functions:'''

Simply put the Richter scale measures the amount of energy released by an earthquake. It translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations in the earthy by an instrument known as the "seismograph". The amplitude of the vibrations or waves tell us how strong aka how much energy was released from the earthquake.

Formula given for the richter scale is:

<math> R = Log\frac{I}{I_o} </math>

which, using your logarithm rules can be written also like (with natural log base of 10:

<math> R = LogA-LogA_o </math>

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-02T00:51:39Z

StephanieUrness:

'''Richter scale and it's relation to logarithmic functions:'''

Simply put the Richter scale measures the amount of energy released by an earthquake. It translates it to a number between 1 and 10, telling us the overall magnitude of the earthquake.

This energy is recorded from resulting vibrations in the earthy by an instrument known as the "seismograph". The amplitude of the vibrations or waves tell us how strong aka how much energy was released from the earthquake.

Formula given for the richter scale is:

<math> R = Log\frac{I}{I_o} </math>

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments). Topics:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13/Logarithmic scale

2011-02-01T20:11:19Z

StephanieUrness: Created page with "'''Richter scale and it's relation to logarithmic functions:'''"

'''Richter scale and it's relation to logarithmic functions:'''

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 13

2011-02-01T18:30:54Z

StephanieUrness:

='''<big><sup>Homework 13</sup></big>'''=

Pick one of the topic offered below and then explain in your own words what it means that these concepts work on a logarithmic scale. Create a wiki page with all your explanations. The length of that page is up to you, but it should feel like the work of four people thinking about a topic and trying to make sense of it. If you drafted something and would like some comments (within 24 hours), send me an email and a link where to look at, I'll post comments).
Topics:
* Decibels
* Richter magnitude scale
* Brightness of stars
* pH

If you have another idea, please send me an email to confirm your choice of topic.

[[/Logarithmic scale/]]

Sandbox:Calculus in Nursing

2011-01-28T23:38:53Z

StephanieUrness: Created page with "Pharmacology: I work on a burn/plastic unit and pain control is, for obvious reasons, an important and never ending issue. Whether it be during dressing changes, physiotherapy ..."

Pharmacology:

I work on a burn/plastic unit and pain control is, for obvious reasons, an important and never ending issue. Whether it be during dressing changes, physiotherapy or daily activities. The question is why can't we give certain combinations of opioids at the same time? Why do we have to wait an allotted amount of time until we

The answer to these questions all involve calculus.

First the concentration of the drug in plasma is measured, this concentration as a function of time, can be plotted and has the ability to tell you several important parameters surrounding dosing, scheduling, metabolism, even absorption. From this you can derive and predict the amount of drug left in the system after "t" amount of time. In the case of different methods of administration (intramuscular, intravenous, orally or subcutaneously, slow release tablets...), the absorption rate will differ, which in the end will affect the amount of time the drug stays in the system. Ie: you may be give a 30mg extended release morphine and instead of it peaking and lasting only d5

User:StephanieUrness

2011-01-26T07:06:28Z

StephanieUrness:

I am taking my first year in my bachelor of science, hoping to major in Environmental sciences. I haven't done any math other than your basics, such as dividing up grocery bills or trying to figure out my height in millimeters (1753mm by the way), for years. This is turning out to be harder than expected!

Analytical Geometry:

When looking up the exact definition for analytical geometry I came across one that summed it up pretty nicely, “analytical geometry is just a fancy name for graphing”. It is geometry (basically about shapes and their properties), using coordinates and algebra. Renee Descartes built on previous methods similar to analytical geometry and was able to use coordinates (numbers describing points) to express geometric relations/shapes in algebraic equations, becoming the “father” of analytical geometry.
Using the most common coordinate system called the Cartesian Coordinate system, where every point has numbers associated with it, one representing the x (horizontal position) coordinate and the other the Y (vertical position) coordinate. Any equation involving coordinates on a plain gives you the solution set for the equation representing the line formed from those points (coordinates). Depending on the equation, the variable and the coordinates, it will tell you the basic picture/type of graph you will get (straight, curved, positive or negative slope). Using these coordinates and numbers you can solve for unknown lengths, angles, vectors, intersecting points… ex: linear equation ax+bx+c = 0, represents a straight line, where a,b, and c are constant numbers, by this you can than plug them into algebraic problems/formulas.
Why is it important? It helped permit the evolution of other modern mathematics such as calculus by Newton and Leibniz (although I do not know who the latter person is).

<big><big>'''Homework 12'''</big></big>

[[Calculus in Nursing]]

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T07:00:07Z

StephanieUrness:

Start with the Function

<math>P(t)=\frac{1}{1+e^{-t}}\quad</math>

1) Change the height of the horizontal asymptote on the right and denote it by K.

2) Change the y-intercept to any number between 0 and K.

*BONUS - Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical.

'''Solution'''

1) We first found where the graph displays horizontal asymptotes. To do this we took the limit of the function as t approaches infinity. This told us as t becomes larger and larger, the function f(x) or "y" becomes arbitrarily close to 1, proving to us we had a horizontal asymptote on the right side. In order to raise this horizontal asymptote on the right side of the graph, we multiplied the whole function by a factor K. Since the numerator is 1, you can simplify and change the graph to:

<math>P(t)=\frac{K}{1+e^{-t}}\quad</math>

K must be equal to or larger than 1, or the horizontal asymptote will lower.

2) To change the y-intercept and to ensure it stays between 0 and K, we shifted the graph to the right. To do this we added any constant h to the the variable t. rewriting the equation:

<math>P(t)=\frac{K}{1+e^{(-t+h)}}\quad</math>

K can be any value from negative INF to positive INF

In the following graph we set K to 3.

Red = original graph

Green = Shifting of horizontal asymptote higher

Blue = Changing the y-intercept

[[File:Shifting of graph to the right.jpg]]

'''BONUS'''

In order to shift the slope of the graph, we must alter the t variable. By adding a coefficient in front of t, you can either increase the slope to almost one (m > 0), or decrease the slope (m < 0).

<math>P(t)=\frac{K}{1+e^{(-mt+h)}}\quad</math>

Red = original equation

Green = Heightened horizontal asymptote on right side and increased slope to near 1
[[File:Slope change.jpg]]

'''Model'''

You can use this function for any type of logistical growth, commonly applied to population growth (whether it be humans, bacteria, animal, or even tumor cells).

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T04:20:27Z

StephanieUrness:

Start with the Function

<math>P(t)=\frac{1}{1+e^{-t}}\quad</math>

1) Change the height of the horizontal asymptote on the right and denote it by K.

2) Change the y-intercept to any number between 0 and K.

*BONUS - Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical.

'''Solution'''

1) We first found where the graph displays horizontal asymptotes. To do this we took the limit of the function as t approaches infinity. This told us as t becomes larger and larger, the function f(x) or "y" becomes arbitrarily close to 1, proving to us we had a horizontal asymptote on the right side. In order to raise this horizontal asymptote on the right side of the graph, we multiplied the whole function by a factor K. Since the numerator is 1, you can simplify and change the graph to:

<math>P(t)=\frac{K}{1+e^{-t}}\quad</math>

K must be equal to or larger than 1, or the horizontal asymptote will lower.

2) To change the y-intercept and to ensure it stays between 0 and K, we shifted the graph to the right. To do this we added any constant h to the the variable t. rewriting the equation:

<math>P(t)=\frac{K}{1+e^{(-t+h)}}\quad</math>

K can be any value from negative INF to positive INF

In the following graph we set K to 3.

Red = original graph

Green = Shifting of horizontal asymptote higher

Blue = Changing the y-intercept

[[File:Shifting of graph to the right.jpg]]

'''BONUS'''

In order to shift the slope of the graph, we must alter the t variable. By adding a coefficient in front of t, you can either increase the slope to almost one (m > 0), or decrease the slope (m < 0).

<math>P(t)=\frac{K}{1+e^{(-mt+h)}}\quad</math>

Red = original equation

Green = Heightened horizontal asymptote on right side and increased slope to near 1
[[File:Slope change.jpg]]

'''Model'''

You can use this function for any type of logistical growth, an area most commonly applied to is populations (whether it be humans, bacteria, animal, or even tumor cells).

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T04:18:41Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T04:17:45Z

StephanieUrness:

File:Slope change.jpg

2011-01-26T04:16:55Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T04:07:50Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T04:07:22Z

StephanieUrness:

Start with the Function

<math>P(t)=\frac{1}{1+e^{-t}}\quad</math>

1) Change the height of the horizontal asymptote on the right and denote it by K.

2) Change the y-intercept to any number between 0 and K.

*BONUS - Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical.

'''Solution'''

1) We first found where the graph displays horizontal asymptotes. To do this we took the limit of the function as t approaches infinity. This told us as t becomes larger and larger, the function f(x) or "y" becomes arbitrarily close to 1, proving to us we had a horizontal asymptote on the right side. In order to raise this horizontal asymptote on the right side of the graph, we multiplied the whole function by a factor K. Since the numerator is 1, you can simplify and change the graph to:

<math>P(t)=\frac{K}{1+e^{-t}}\quad</math>

K must be equal to or larger than 1, or the horizontal asymptote will lower.

2) To change the y-intercept and to ensure it stays between 0 and K, we shifted the graph to the right. To do this we added any constant k to the the variable t. rewriting the equation:

<math>P(t)=\frac{K}{1+e^{(-t+K)}}\quad</math>

K can be any value from negative INF to positive INF

In the following graph we set K to 3.

Red = original graph

Green = Shifting of horizontal asymptote higher

Blue = Changing the y-intercept

[[File:Shifting of graph to the right.jpg]]

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T04:00:19Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T03:53:27Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T03:53:03Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T03:48:21Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve

2011-01-26T03:46:43Z

StephanieUrness:

{{
Infobox MATH110 Teams
| team name = Geneve
| member 1 = Agnes Luong
| member 2 = Deborah Ma
| member 3 = Megan Bontogon
| member 4 = Stephanie Urness
}}
In workshop J.

'''<big><big>Homework 11</big></big>'''

[[/Course:Math110/003/Teams/Geneve/Homework11/]]

<big><big>'''Homework 12'''</big></big>

[[Course:MATH110/003/Teams/Geneve/Homework_12]]

'''<big><big>Word of the day</big></big>'''

Agnes -

Deborah - ''Motivation''

Megan -

Stephanie -

Course:MATH110/Archive/2010-2011/003/Teams/Geneve

2011-01-26T03:46:23Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T03:44:36Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T03:34:01Z

StephanieUrness:

File:Shifting of graph to the right.jpg

2011-01-26T03:27:50Z

StephanieUrness: uploaded a new version of "File:Shifting of graph to the right.jpg"

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T03:24:02Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T03:21:59Z

StephanieUrness:

Start with the Function

math>P(t)=\frac{1}{1+e^{-t}}\quad</math>

1) Change the height of the horizontal asymptote on the right and denote it by K.
2) Change the y-intercept to any number between 0 and K.
*BONUS - Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical.

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T03:21:08Z

StephanieUrness: /* Homework 12 */

File:Shifting of graph to the right.jpg

2011-01-26T03:09:45Z

StephanieUrness:

File:Graph of horizontal A.jpg

2011-01-26T03:04:56Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Teams/Geneve/Homework 12

2011-01-26T02:59:04Z

StephanieUrness:

Start with the function
:<math>P(t) = \frac{1}{1+e^{-t}}</math>

1) Change the height of the horizontal asymptote on the right, denote it by <math>K</math>.

2) Change the <math>y</math>-intercept to any number between <math>0</math> and <math>K</math>

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T08:16:31Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

A simple example is the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at x = -3, we must redefine this point (technically the graph should have an open circle at x=-3, in order to make this functions continuous

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

The graph for this function will look exactly the same however where there was a circle with a hole at x = -3 there will now be a filled in circle to indicate this point is defined. It is now continuous.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. The function is UNDEFINED at x = -3, you can say the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:

[[File:step functions example resized.jpg]]

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

== References ==

1) http://www.mathwords.com/r/removable_discontinuity.htm

2) http://www.brightstorm.com/math/algebra-2/additional-topics/piecewise-functions

3) http://patrickjmt.com/

4) http://kerbaugh.uncfsu.edu/piecewise/piecewise.html

5)http://mathdemos.gcsu.edu/mathdemos/piecewise/piecewise_differentiability.html

6) http://mathforum.org/dr/math/

7) http://www.wolframalpha.com/

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T08:14:21Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

A simple example is the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at x = -3, we must redefine this point (technically the graph should have an open circle at x=-3, in order to make this functions continuous

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

The graph for this function will look exactly the same however where there was a circle with a hole at x = -3 there will now be a filled in circle to indicate this point is defined. It is now continuous.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:

[[File:step functions example resized.jpg]]

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

== References ==

1) http://www.mathwords.com/r/removable_discontinuity.htm

2) http://www.brightstorm.com/math/algebra-2/additional-topics/piecewise-functions

3) http://patrickjmt.com/

4) http://kerbaugh.uncfsu.edu/piecewise/piecewise.html

5)http://mathdemos.gcsu.edu/mathdemos/piecewise/piecewise_differentiability.html

6) http://mathforum.org/dr/math/

7) http://www.wolframalpha.com/

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T08:10:12Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

A simple example is the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at -3, we must redefine this point (technically the graph should have an open circle at x=-3. We see where that point is and redefine x to equal the "y" value in order to make continuous.

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

The graph for this function will look exactly the same however where there was a circle with a hole at x = -3 there will now be a filled in circle to indicate this point is defined. It is now continuous.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:

[[File:step functions example resized.jpg]]

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

== References ==

1) http://www.mathwords.com/r/removable_discontinuity.htm

2) http://www.brightstorm.com/math/algebra-2/additional-topics/piecewise-functions

3) http://patrickjmt.com/

4) http://kerbaugh.uncfsu.edu/piecewise/piecewise.html

5)http://mathdemos.gcsu.edu/mathdemos/piecewise/piecewise_differentiability.html

6) http://mathforum.org/dr/math/

7) http://www.wolframalpha.com/

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T08:09:53Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

A simple example is perhaps the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at -3, we must redefine this point (technically the graph should have an open circle at x=-3. We see where that point is and redefine x to equal the "y" value in order to make continuous.

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

The graph for this function will look exactly the same however where there was a circle with a hole at x = -3 there will now be a filled in circle to indicate this point is defined. It is now continuous.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:

[[File:step functions example resized.jpg]]

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

== References ==

1) http://www.mathwords.com/r/removable_discontinuity.htm

2) http://www.brightstorm.com/math/algebra-2/additional-topics/piecewise-functions

3) http://patrickjmt.com/

4) http://kerbaugh.uncfsu.edu/piecewise/piecewise.html

5)http://mathdemos.gcsu.edu/mathdemos/piecewise/piecewise_differentiability.html

6) http://mathforum.org/dr/math/

7) http://www.wolframalpha.com/

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T05:41:33Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

For example:

A simple example is perhaps the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at -3, we must redefine this point (technically the graph should have an open circle at x=-3. We see where that point is and redefine x to equal the "y" value in order to make continuous.

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

The graph for this function will look exactly the same however where there was a circle with a hole at x = -3 there will now be a filled in circle to indicate this point is defined. It is now continuous.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:
[[File:step functions example..jpg]]

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

== References ==

1) http://www.mathwords.com/r/removable_discontinuity.htm

2) http://www.brightstorm.com/math/algebra-2/additional-topics/piecewise-functions

3) http://patrickjmt.com/

4) http://kerbaugh.uncfsu.edu/piecewise/piecewise.html

5)http://mathdemos.gcsu.edu/mathdemos/piecewise/piecewise_differentiability.html

6) http://mathforum.org/dr/math/

7) http://www.wolframalpha.com/

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T05:41:22Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

For example:

A simple example is perhaps the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at -3, we must redefine this point (technically the graph should have an open circle at x=-3. We see where that point is and redefine x to equal the "y" value in order to make continuous.

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

The graph for this function will look exactly the same however where there was a circle with a hole at x = -3 there will now be a filled in circle to indicate this point is defined. It is now continuous.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:
[[File:step functions example..jpg]]

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

== References ==

1) http://www.mathwords.com/r/removable_discontinuity.htm

2) http://www.brightstorm.com/math/algebra-2/additional-topics/piecewise-functions

3) http://patrickjmt.com/

4) http://kerbaugh.uncfsu.edu/piecewise/piecewise.html

5)http://mathdemos.gcsu.edu/mathdemos/piecewise/piecewise_differentiability.html

6) http://mathforum.org/dr/math/

7) http://www.wolframalpha.com/

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T05:40:47Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

For example:

A simple example is perhaps the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at -3, we must redefine this point (technically the graph should have an open circle at x=-3. We see where that point is and redefine x to equal the "y" value in order to make continuous.

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

The graph for this function will look exactly the same however where there was a circle with a hole at x = -3 there will now be a filled in circle to indicate this point is defined. It is now continuous.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:
[[File:step functions example..jpg]]

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

== References ==

1) http://www.mathwords.com/r/removable_discontinuity.htm
2) http://www.brightstorm.com/math/algebra-2/additional-topics/piecewise-functions
3) http://patrickjmt.com/
4) http://kerbaugh.uncfsu.edu/piecewise/piecewise.html
5)http://mathdemos.gcsu.edu/mathdemos/piecewise/piecewise_differentiability.html
6) http://mathforum.org/dr/math/
7) http://www.wolframalpha.com/

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T04:20:21Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

For example:

A simple example is perhaps the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at -3, we must redefine this point (technically the graph should have an open circle at x=-3. We see where that point is and redefine x to equal the "y" value in order to make continuous.

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

The graph for this function will look exactly the same however where there was a circle with a hole at x = -3 there will now be a filled in circle to indicate this point is defined. It is now continuous.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T04:19:29Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

For example:

A simple example is perhaps the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at -3, we must redefine this point (technically the graph should have an open circle at x=-3. We see where that point is and redefine x to equal the "y" value in order to make continuous.

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

The graph for this function will look exactly the same however where there was a circle with a hole at x = -3 there will now be a filled in circle to indicate this point is defined.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T04:16:59Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

For example:

A simple example is perhaps the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

[[File:JumpD.gif]]

Since x is NOT defined at -3, we must redefine this point (technically the graph should have an open circle at x=-3. We see where that point is and redefine x to equal the "y" value in order to make continuous.

So our function, after redefining, will look like:

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \\ 4 & x \end{cases}</math>

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

File:JumpD.gif

2010-12-03T04:11:27Z

StephanieUrness:

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T04:10:47Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

For example:

A simple example is perhaps the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous.

Say you have a function

<math>f(x) = \begin{cases} x+7 & x<-3 \\ x((8/3)-4) & x>-3 \end{cases}</math>

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}

Course:MATH110/Archive/2010-2011/003/Groups/Group 06/Basic Skills - Piecewise Functions

2010-12-03T03:59:41Z

StephanieUrness:

== '''What is it?''' ==

A piecewise function is a function which is broken up into different pieces, each piece is defined on a certain interval. These intervals depend on the independent variable (x) and define the function.

Regular functions usually apply the same process no matter which number it is given, whereas a piecewise function will look at the number first and based on that number itself and where it is found, will decide which formula to put that number into.

For example, this is a piecewise function

<math>f(x) = \begin{cases} 5x+2& x \leq 0 \\ x+2 & x>0 \end{cases}</math>

So the "formula" 5x+2 is only used when your independent variable "x" is smaller or equal to 0. If the value of "x" is any greater than 0 you must use the "formula" x+2 to find the value of your function at that point. These two pieces define the whole function.

[[File:1examplepiece.jpg]]

Piecewise functions are useful for graphing curves that differentiate over their domains, such such as in economic models where major factors (ie. factors of production) are altered over the changing domain.

== '''How to Graph them''' ==

A piecewise function is graphed by graphing each of the "pieces" over the function's specific domain, as defined by an inequality or interval statement.

An easy way to do this is to graph each of the functions on your plot, and then erase portions of each function based on the domain statements.

For example:

A simple example is perhaps the absolute value of f(x)= |x|, we can break this function up into two pieces (Yes a piecewise function!).

<math>f(x) = \begin{cases} -x & x \leq 1 \\ x & x>0 \end{cases}</math>

Look at the domain restriction of each "piece" and simply draw each of the functions on a graphing plane and then erase the part of the function that is out of the range of its specific inequality.

[[File:Absolutevalue.jpg]]

Most piecewise functions you come across will not look this simple, lets look at a slightly more complex looking piecewise function:

<math>f(x) = \begin{cases} x^2+1 & x \leq 4 \\ 6 & x>4 \end{cases}</math>

It may be easiest, and a good habit, to start off by drawing vertical dotted lines on the graph where your dividing line(s) is/are. For this example their is only one and it would be "4" (you can tell by looking at the restricting domains). Draw each of the functions on a graph until that restricting line, if that portion of the graph includes that point (<math>x\leq</math>) draw a circle which is filled in, if not <math> x>1 </math> than leave the inside of the circle uncolored.

So, for F(x) = x^2+1, erase any part of the function that appears to the right of x=4. You are doing this because x^2+1 is only the valid function for any values of x that are smaller than 4.

[[File:Piecewisegraphing.jpg]]

=== Video ===

Here is an awesome video giving a very clear demonstration on how to graph piecewise functions when x is defined between 2 values,
ie: <math>-2<x\leq5</math> .
This video points out the importance of knowing what each piece looks like before graphing the whole function.

{{#ev:youtube | -gwffMEr8i8 | 400}}

== ''' Continuity in Regards to Piecewise Functions ''' ==

In order to analyze if a function of "f" is continuous at a point one must look and see if the point satisfies the following, if a=x :

1. "a" is in the domain of f which means it is defined everywhere in the function

2. constitute functions of f are continuous throughout the function

3. discontinuity doesn't exist at the end points of each functions intervals.

Basically if a graph is continuous it will have no holes or breaks, however it may twist, turn and change direction (each of these are pieces). A good basic rule is: If you can draw the whole graph without lifting your pen from the paper it is continuous.

Is the piecewise function below continuous?

Piecewise functions are made up of a finite number of continuous pieces. One taboo for continuous piecewise functions are vertical asymptotes, they cannot be continuous when they have vertical asymptotes. Although it is possible for a continuous piecewise function to have removable and step discontinuities. Why? Because you can redefine x at this point, and in doing this you make the function continuous. With a vertical asymptote you cannot redefine x to make the asymptote disappear as you can with the other discontinuities (See below for examples).

== '''How to modify them?''' ==

=== Modifying to make continuous ===

We have talked about piecewise functions which only have x and y as "unknown" variables. What about situations when their is another variable thrown in to represent a coefficient which can make the function continuous if the right value is plugged in.

Say you have a function:

<math>f(x) = \begin{cases} x^2+5 & x \leq 1 \\ \frac{2x+a}{x+2} & x>1 \end{cases}</math>

Depending on the value of a, the two graphs may or may not join together. What if we want it to be continuous? What value must "a" be for this function to be continuous (remember this does not mean it is differentiable).

For the function to be continuous the graphs from each "piece" must connect, we need to find the value of "a" which will make the "pieces" connect. Looking at the defined domains of each piece, you can see the point at which they break up is when x = 1. This is the part of the graph we want to focus our attention on and find a value which will make the two pieces meet at this point.

If we put 1 into the first piece <math>f(1)= (1)^2+5 =6 </math>

The point at which this piece stops is at (1,6). For the function to be continuous the second piece must start at this point (1,6).
So when x=1, the second piece must equal 6 <math>f(1)= \frac{2(1)+a}{(1)+2}=6</math>

Solving for "a" we get a = 16.

In order for this piecewise function to be continuous "a" must equal 16.

[[File:Modfyjumpdiscontinuity.gif]]

Is 16 the only value for a which makes the function continuous? Lets make a = 4.

[[File:Xis4.gif]]

You can see there is a break in the graph at x = 1, therefore it is not continuous. You can plug in any other value into a and it will give you a graph, however it will not be continuous.

=== Removable discontinuity ===

Why is it above we described a piecewise continuous function as one which can have a removable discontinuity? This is because if we modify it slightly we can we can make it continuous. All you need to do is redefine the point at which it is discontinuous, and make it continuous.

=== Piecewise function with an asymptote ===

<math>f(x) = \begin{cases} \frac{x^2+7}{x+3}& x\leq 1 \\ \frac{2x+a}{3x+4} & x>1 \end{cases}</math>

Can we make this continuous? Lets try, what value must "a" be for this function to be "continuous".

Remember we look at the defined domains of each piece, the point at which they break up is when x = 1.

If we put 1 into the first piece <math>f(1)= \frac{(1)^2+7}{(1)+3} =2</math>

The point at which this piece stops is at (1,2). For the function to be continuous the second piece must start at this point (1,2).
So when x=1, the second piece must equal 2 <math>f(1)= \frac{2(1)+a}{3(1)+4}=2</math>

Solving for a we get a=12.

In order for this piecewise function to be continuous "a" must = 12. The graph below shows the continuity of the function when a = 12, notice the vertical asymptote.

[[File:Equal12.jpg]]

The vertical asymptote is a point on the graph where x approaches but never reaches (from left and right side). Can you draw this graph without lifting your pen? No, therefore the function cannot be defined as continuous. However you can say -3 is undefined, and the graph is continuous everywhere but where it is undefined.

== ''' Differentiability in Regards to Piecewise Functions ''' ==

In graphing this function we realize it is not differentiable, because when you take the derivative the two pieces do not match.
In order for a function to be differentiable it must satisfy two conditions:
1) Be continuous
2) The "pieces" must match with the same slope

What does this mean?

== ''' Different Types of Step Functions ''' ==

Step Functions: are functions that look like steps when plotted on a graph and they follow the same piecewise rules. They are also called linear piecewise function graphs. The reason behind their correlation with piecewise functions are they share the same idea of small line segments.

Below is an example:

== '''How to use them?''' ==

Piecewise functions are useful because they can show a situation where the graph changes radically and to a point outside of the equations range, as often happens in real life.

Take, for example, when you sell tickets to the opera. If someone wanted to create a strict representation of total revenues earned against number of tickets sold, they would have to use a piecewise graph. Tickets cannot be sold in parts, as only one person will occupy one seat at a time; It would be impractical to sell 1.5 tickets, for example. Because you can't sell portions of a ticket the graph technically cannot be linear, it must have interruptions after each integer. When you are selling tickets to the opera, the rate of change for the graph between whole integers is 0 (each section of the graph between integers is flat). So, each ticket sold would have its own function defining it, specifically in the form of y=Px, x=x

== ''' Links ''' ==

Here are a couple great videos explaining and demonstrating piecewise functions. (http://patrickjmt.com/)

This video gives a great overview of basic piecewise functions and graphing. In general a few examples on how to evaluate piecewise functions with different values of x (ex. f(-4) and f(2)), and than graph them depending on which interval that number is defined.

{{#ev:youtube | hy0N-90gCu0 | 400}}

Finding Domain and Range of Piecewise Functions:

{{#ev:youtube | BxaYyS6lsQ4 | 400}}

Find the formula for a Piecewise Function from a Graph

{{#ev:youtube | eWo8tWuaGfU | 400}}