Science:Math Exam Resources/Courses/MATH110/December 2015/Question 01 (b)

MATH110 December 2015

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Question 01 (b)
Consider a function $f$ such that $f(x)<0$ for $x>0$ and $f(x)>0$ for $x<0$. Let $g(x)=x^{2}$. Determine the sign of the composite function $f(g(x))$ if $x>0$ and if $x<0$.

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 sign of $g(x)$? How would the sign of $g(x)$ help us determine the sign of $f(x)$?

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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. Take $g(x)=x^{2}=u$, since this is a square function it is always nonnegative, i.e. whether $x>0$ or $x<0$, $g(x)=u>0$. Thus the composite function $f(g(x))=f(u)$ is negative i.e. $f(u)<0$ because $u>0$. Answer: In both case, $\color {blue}f(g(x))<0$.