Let’s take a function h(x), whose domain is the real numbers. We are simply going to start by writing h(x) in a slightly strange way. We will write it as:

$h(x)=\dfrac{h(x)+h(-x)+h(x)-h(-x)}{2}$

This might seem an odd thing to do – we have essentially added zero to the original function (in the form h(-x)-h(-x)). However, we can see that we can split this as:

$h(x)=\dfrac{h(x)+h(-x)}{2}+\dfrac{h(x)-h(-x)}{2}$

It’s exactly the same thing we started with, right? But now it’s written in a peculiar way. Now let’s call the two fractions f(x) and g(x) respectively. So:

$f(x)=\dfrac{h(x)+h(-x)}{2}$

and

$g(x)=\dfrac{h(x)-h(-x)}{2}$

So our original function can be written as h(x)=f(x)+g(x). If you plug in f(x) and g(x) above you will see that we have said nothing which is not trivial in any of this. However, the interesting part comes when we look at the properties of f(x) and g(x). What is f(-x)?

$f(-x)=\dfrac{h(-x)+h(x)}{2}=\dfrac{h(x)+h(-x)}{2}=f(x)$

But this is just the defining property of an even function, so f(x) is even. How about g(-x)?

$g(-x)=\dfrac{h(-x)-h(x)}{2}=-\dfrac{h(x)-h(-x)}{2}=-g(x)$

But this is just the defining property of an odd function, so g(x) is odd.

How does this help us? Well, let’s say you are given a function like h(x)=|x-32| and asked to write to write this as the sum of an odd and an even function. Well, the even part is given by:

$\dfrac{h(x)+h(-x)}{2}=\dfrac{|x-32|+|-x-32|}{2}=\dfrac{|x-32|+|x+32|}{2}$

and the odd part is:

$\dfrac{h(x)-h(-x)}{2}=\dfrac{|x-32|-|-x-32|}{2}=\dfrac{|x-32|-|x+32|}{2}$

These two are odd and even respectively, and you can see that when you add them together, they give you the original function. Easy as that.

 How clear is this post?