1. ## Crazy Calculus Project

This is my crazy calculus professor coming up with this stuff.

He wants us to show that if f is any function, then it can be written as the sum of even and odd functions. That is, if f[even] is an even function and
f[odd] is an odd function, then:
f(x) = f[even](x) + f[odd](x)
He gives an example.
Example:
For f(x) = 1/(1+x), we can take f[even](x)=1/(1-x^2) and f[odd](x)=-x/(1-x^2).
You can check for yourself that f[even](x) is in fact even and that f[odd](x) is indeed odd. And also you can see that:

f[even]+f[odd]=[1/(1-x^2)] +[-x/(1-x^2)]=(1-x)/(1-x^2)=1/(1+x)=f(x)

Even with this example, I am still confused as to how to show....(see paragraph one)

2. Consider these functions.
$\displaystyle g(x) = \frac{{f(x) + f( - x)}}{2}\quad \& \quad h(x) = \frac{{f(x) - f( - x)}}{2}$

What if i said:
Say f(x)=(x^3+2x^2-x-1)/(x^2-4)
let g(x)=f[even](x)=(2x^2-1)/(x^2-4)
and let h(x)=f[odd](x)=(x^3+x)/(x^2-4)
Then g(x)+h(x)=f(x) am I right?

4. Do you understand that your professor is asking you to show:
Any function f can be written as the sum of an even function and an odd function?
$\displaystyle g(x) = \frac{{f(x) + f( - x)}}{2}\quad \Rightarrow \quad g(x) = g( - x)\mbox{ even}$
$\displaystyle h(x) = \frac{{f(x) - f( - x)}}{2}\quad \Rightarrow \quad h(x) = - h( - x)\mbox{ odd}$
$\displaystyle f(x) = g(x) + h(x)$

Now that is true for Any function f.