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

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.

• Do not use Fourier series. Given any $f(x)$, find $g(x)$ and $h(x)$ in terms of $f(x)$ so that $f(x) = g(x) + h(x)$. For example $g(x) = e^{f(x)} + e^{f(-x)}$ is even, but of course you can make a much simpler function $g$ that is even. Oct 21, 2019 at 16:09
• Can you tell which terms of your $f(x)$ are even and which are odd? Group them accordingly.
– MPW
Oct 21, 2019 at 16:10
• @MPW x and x^3 are odd and x^2 is even Oct 21, 2019 at 16:12
• So doesn't that mean that $f(x)=\underbrace{(x^2)}_{\textrm{even}} + \underbrace{(x+x^3)}_{\textrm{odd}}$ fits the bill?
– MPW
Oct 21, 2019 at 16:14
• @MPW now i got it...f(x)=h(x)+g(x),. Where h(x)=x^2 is even and g(x)=x+x^3 is odd....thanks Oct 21, 2019 at 16:24

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 $$g(x)=...$$