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. :))

Hint
Just think this way: What property does a function obey so that you can call it even? $g(x)=g(-x)$. And similarly $g(x)=-g(-x)$ for the odd one. Now verify that $g(x)=f(x)+f(-x)$ is even(What is $g(x)$ and what is $g(-x)$? Do you find them equal?) .Similarly for the second one.
