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. :))

$$f(-x) + f(-(-x))=f(-x)+f(x)=f(x)+f(-x)$$
$$-[f(-x) - f(-(-x))]=-[f(-x) - f(x)]=-f(-x)+f(x)=f(x)-f(-x)$$
