Science:Math Exam Resources/Courses/MATH110/December 2012/Question 01 (b)

MATH110 December 2012

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Question 01 (b)
Sketch the graph of the following function. $\displaystyle g(x)=\ln(1-x)$

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 for the usual logarithm, $f(x)=\ln(x)$, x can't be zero or negative. Which values of x will make sure that the above function doesn't end up with a negative or zero value inside the logarithm?

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1
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 graph of $g(x)=\ln x$ looks like this: Using function rules for the graph of $g(x)=\ln(1-x)$ the negative in front of the x means the graph has been reflected over the y-axis and (1-x) inside the function means that it is shifted right 1 giving:

Solution 2

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. If you've forgotten the rules used in the first solution (or never learned them), you can also facts about the domain of $\ln(x)$ to determine where the graph should go. The domain of the function $f(x)=\ln(x)$ is $x>0$ which can be seen in the graph below. The graph of $g(x)=\ln(1-x)$ will have a similar shape, but has a different domain. Let's test some values to see where the domain of this function is. x-value y-value -2 $\ln(1-(-2))=\ln(3)$ $\ln(3)$ -1 $\ln(1-(-2))=\ln(2)$ $\ln(2)$ 0 $\ln(1-0)=\ln(1)$ 0 1 $\ln(1-1)=\ln(0)$ undefined 2 $\ln(1-2)=\ln(-1)$ undefined So clearly this function is defined on the domain $x<1$, suggesting the graph looks something like this:

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