How to determine if -h(-x) is even odd or neither

Question

I know that h(x) is even so it definitely can't be A, so it could be one of the other 3 choices, but I am not sure how to determine if -h(-x) is even odd or neither.

Answer 1

You're right that $h(x)$ is even, which we can see by evaluating $h(-x)$ as follows. $$h(-x) = g(f(-x)) = g(-f(x)) = g(f(x)) = h(x)$$

Let $g(x) = -h(-x)$. We want to see whether $g$ is even, so let's evaluate $g(-x)$. $$g(-x) = -h(-(-x)) = -h(x) \stackrel{*}{=} -h(-x) = g(x)$$

*We used the fact that $h(x)$ is even to turn $h(x)$ into $h(-x)$.

Therefore, $g(x)$ is also even. You should now be able to determine the right answer.