Science:Math Exam Resources/Courses/MATH 180/December 2017/Question 11 (a)

MATH 180 December 2017

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Question 11 (a)
The function $$P(x)={\frac {1}{4}}xe^{-x/2},\quad x\geq 0,$$ is an example of a Gamma distribution, a function that may be used to describe certain probabilities. (a) Write down the coordinates of all ${\textstyle x}$- and ${\textstyle y}$-intercepts of ${\textstyle P(x)}$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Recall that the exponential function is always positive.

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The ${\textstyle y}$-intercept is $$P(0)={\dfrac {1}{4}}\cdot 0\cdot e^{-0/2}=0.$$ The ${\textstyle x}$-intercept is given by solving $$P(x)={\dfrac {1}{4}}xe^{-x/2}=0.$$ By Hint 1, the exponential function is positive. Throwing away the positive factors, we are left with $$x=0.$$ That means the only ${\textstyle x}$-intercept is ${\textstyle 0}$. Answer: the only ${\textstyle x}$-intercept is ${\textstyle {\color {blue}0}}$ and the ${\textstyle y}$-intercept is ${\textstyle {\color {blue}0}}$.