Proving any function can be written in the form f = O + E ?
If $\displaystyle E$ is any function such that:
$\displaystyle E(x) = E(-x) $
And $\displaystyle O$ is any function such that:
$\displaystyle -O(x) = O(-x)$
Then prove that any given function $\displaystyle f$ can be expressed such that:
$\displaystyle f(x) = E(x) + O(x)$
Now, I will show what work I've done and see if anybody can tell me if I'm on the right track. Lets assume firstly that $\displaystyle f$ is even. Let $\displaystyle h$ be any choosen odd function and $\displaystyle g$ be any choosen even function; then we can set:
$\displaystyle f = g$
Then we know that f can be written as:
$\displaystyle f(x) = g(x) = f(-x) = g(-x)$
Now, if we let:
$\displaystyle h(x) = 0$
then:
$\displaystyle h(x) = h(-x) = -h(x)$
So $\displaystyle h$ is either even or odd, so if $\displaystyle f$ is even we can write it as:
$\displaystyle f(x) = g(x) + h(x) = g(x) + 0 = g(x) = f(x)$
Assuming $\displaystyle f$ is odd then we can set:
$\displaystyle f(x) = h(x)$
And simmiliarly to the above:
$\displaystyle g(x) = 0$
So we know g can be either even or odd. So we let g stand in for our even function, and we can write f as:
$\displaystyle f(x) = h(x) + g(x) = h(x) + 0 = h(x) = f(x)$
But, how do I now prove the cases where we assume $\displaystyle f(x) = 0$ and when $\displaystyle f$ is not even or odd? Any guidance would be appreciated. Thanks
