Science:Math Exam Resources/Courses/MATH110/April 2011/Question 08 (c)

MATH110 April 2011

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Question 08 (c)
Let $\displaystyle f(x)=\ln \left(4-x^{2}\right)$ Find all the x- and y-intercepts of ƒ.

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
What is the value of $x$ at a y-intercept? How can you use this to find $y$? Similarly, what is the value of $y$ at an x-intercept?

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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. At a function's x-intercept, its y-value is equal to zero. Thus to find our x-intercepts, we need to set the function equal to 0 and solve for $x$. $0=\ln(4-x^{2})$ We raise both sides by e to cancel the logarithm and get: ${\begin{aligned}e^{0}&=e^{\ln(4-x^{2})}\\1&=4-x^{2}\\x^{2}&=3\\\pm {\sqrt {3}}&=x\end{aligned}}$ So our two x-intercepts are at $x=\pm {\sqrt {3}}$. The y-intercept of a function occurs when $x=0$. Therefore the y-intercept is $\displaystyle {}y=\ln {4}$.

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