Constant raised to the power of an even or odd function

Question

Suppose that $a$ is a positive real number, that $f(x)$ is an even function and that $g(x)$ is an odd function.

Would $a^{f(x)}$ be an even or odd function?

And would $a^{g(x)}$ be an even or odd function?

Please explain why :)

Answer 1

This is one of those problems in which all you have to do is apply the definitions.

Since $f$ is even, $f(-x) = f(x)$. Since $g$ is odd, $g(-x) = -g(x)$.

Then, $a^{f(-x)} = a^{f(x)}$. Based on this, is $a^{f(x)}$ even or odd or neither?

Also, $a^{g(-x)} = a^{-g(x)} = \dfrac{1}{a^{g(x)}}$. Based on this, is $a^{g(x)}$ even or odd or neither?

Note: The answer to this last one changes if $a = 1$.