Suppose f(x) is an odd function. Prove that g(x) = |f(x)| is an even function.

Question

Suppose f(x) is an odd function. Prove that g(x) = |f(x)| is an even function. I understand that an odd function is where f(-x) = -f(x), and an even function is where f(-x) = f(x), but am struggling with actually proving the question.

Answer 1

$g(-x) = |f(-x)| = |-f(x)| = |f(x)| = g(x)$. So $g$ is even.

Answer 2

You've nearly answered your own question in the OP. :)

We know $f$ is an odd function, so $f(-x) = -f(x)$. Thus we have $|f(-x)| = |-f(x)| = f(x)$.

In particular, notice that $|f(-x)| = |f(x)|$. We conclude that $|f(x)|$ is an even function.