# Proving any function can be written in the form f = O + E ?

• Oct 5th 2010, 12:34 PM
mfetch22
Proving any function can be written in the form f = O + E ?
If $E$ is any function such that:

$E(x) = E(-x)$

And $O$ is any function such that:

$-O(x) = O(-x)$

Then prove that any given function $f$ can be expressed such that:

$f(x) = E(x) + O(x)$

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 $f$ is even. Let $h$ be any choosen odd function and $g$ be any choosen even function; then we can set:

$f = g$

Then we know that f can be written as:

$f(x) = g(x) = f(-x) = g(-x)$

Now, if we let:

$h(x) = 0$

then:

$h(x) = h(-x) = -h(x)$

So $h$ is either even or odd, so if $f$ is even we can write it as:

$f(x) = g(x) + h(x) = g(x) + 0 = g(x) = f(x)$

Assuming $f$ is odd then we can set:

$f(x) = h(x)$

And simmiliarly to the above:

$g(x) = 0$

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:

$f(x) = h(x) + g(x) = h(x) + 0 = h(x) = f(x)$

But, how do I now prove the cases where we assume $f(x) = 0$ and when $f$ is not even or odd? Any guidance would be appreciated. Thanks
• Oct 5th 2010, 12:48 PM
Defunkt
Separation into cases is not needed --
You can write $\displaystyle f(x) = \frac{f(x)+f(-x) -f(-x)+f(x)}{2} = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}$
• Oct 5th 2010, 03:52 PM
mfetch22
Quote:

Originally Posted by Defunkt
Separation into cases is not needed --
You can write $\displaystyle f(x) = \frac{f(x)+f(-x) -f(-x)+f(x)}{2} = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2}$

Wait, did u assume $f$ to be even or odd in the above work? Or did you simply manipulate it into an equal form which represents the ability to equate the "even and odd" forms of $f$?
• Oct 6th 2010, 02:29 AM
Defunkt
Quote:

Originally Posted by mfetch22
Wait, did u assume $f$ to be even or odd in the above work? no Or did you simply manipulate it into an equal form which represents the ability to equate the "even and odd" forms of $f$? correct

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