Proof in calculus Prove that, if ${f(x)}$ is any function, ${f(x) + f(-x)}$ is an even function while ${f(x) - f(-x)}$ is an odd function.
Thank you!
Note: I have used this theorem a lot of time. And I can prove it by taking specific functions. But, I have no idea about how to prove it for a general function ${f(x)}$. Even if someone can give me hints about how I should proceed, it will be really helpful. :))

Proof:
Let $g(x) = f(x) + f(-x)$:
Then, $g(-x) = f(-x) + f(x) = g(x)$ so it is even
Let $h(x) = f(x) - f(-x)$:
Then, $-h(-x) = -f(-x) + f(x) = h(x)$ so it is odd
