Ok im taking an analysis course and im having trouble with one of these proofs.(adsbygoogle = window.adsbygoogle || []).push({});

Prove any function f(x) can be written uniquely as f(x) = E(x) + O(x) when E is and even function and O is an odd function.

So to try and prove it i did this:

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

E(-x) = E(x) since E is an even function

O(-x) = -O(x) since O is an odd function

therefore

E(-x) + O(-x) = E(x) + (-O(x)) = E(x) - O(x)

Im sure that is right i just don't see how this could be a proof that any function f(x) can be written uniquely as f(x) = E(x) + O(x) when E is and even function and O is an odd function.

So i did some more calculations:

Suppose E and O are are both even functions, then:

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

Suppose E and O are both odd functions, then:

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

and finally suppose E is odd and O is even:

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

Am on on the right track of did i completely miss something?

Thanks

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# Proof help

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