Answer
Only the function $f(x)=0$ can be both odd and even.
Work Step by Step
Let's have a random function called $f(x)$.
- $f(x)$ is an even function when $f(-x)=f(x)$
- $f(x)$ is an odd function when $f(-x)=-f(x)$
So if $f(x)$ wants to be both even and odd, that means
$$f(-x)=f(x)=-f(x)$$
We focus only on this part: $$f(x)=-f(x)$$
$$2f(x)=0$$
$$f(x)=0$$
That means $f(x)=0$ is both odd and even, which makes sense when we consider: $f(-x)=0$ and $-f(x)=0$.
