How can a function defined on symmetrically placed interval be written as sum of an even and odd function?

Question

I know how to find Fourier series of a function,but i found following question and I stuck.

"Show that any function $f(x)$ defined on symmetrically placed intervals can be written as sum of an even function and an odd function .

Hence show that how to write $f(x) =x+x^2+x^3$ as per above statement.

Can anyone please help me...thanks in advance.

Answer 1

Suppose $f(x)=h(x)+g(x)$ where $h(x)$ is even, $g(x)$ is odd. Then $f(-x)=h(x)-g(x)$. Adding both equations we get that $h(x)=\frac{f(x)+f(-x)}{2}$ and $g(x)=...$