# Thread: odd and even functions proof

1. ## odd and even functions proof

Prove that every function de fined on the real line can be written as a sum of an even and an odd
function.

I'm not really sure where to start - any hints would be much appreciated

2. ## Re: odd and even functions proof

Hey kinhew93.

I'd say the first thing to do is write down the definition of an odd ad even function. It is asking to show that f(x) = g(x) + h(x) holds for even functions g(x), odd functions h(x) and even/odd functions f(x).

Hint: Try using the fact that f(x)/g(x) = 0 is an even/odd function as a first step.

3. ## Re: odd and even functions proof

Big hint:

the function:

g(x) = f(x) + f(-x) is an even function, for any function f.

4. ## Re: odd and even functions proof

And h(x)= f(x)- f(-x) is an odd function, for any function f!

5. ## Re: odd and even functions proof

Originally Posted by Deveno
g(x) = f(x) + f(-x) is an even function, for any function f.
Originally Posted by HallsofIvy
And h(x)= f(x)- f(-x) is an odd function, for any function f.
Is it true that $f(x)=\frac{g(x)}{2}+\frac{h(x)}{2}~?$