An

odd function satisfies f(x) = -f(-x) and

even functions satisfy f(x) = f(-x).

These kinds of functions are very useful

* because of their

inherent symmetry.

Amazingly any old

non-symmetric,

weird,

common or

garden function can be represented as the

sum of an

odd function and an

even function.

**Functions which always work in the above:**
Let f(x) be an arbitrary function.
Now, assume that there exists an even function fe(x) and an odd function fo(x) such that
f(x) = f_{e}(x) + f_{o}(x)
*even* *odd*
*then*
f(-x) = f_{e}(-x) + f_{o}(-x) = f_{e}(x) - f_{o}(x)
Solving continuously gives:
f_{e}(x) = 1/2 (f(x) + f(-x))
*and*
f_{o}(t) = 1/2 (f(x) – f(-x)) **!**

***Why is this useful?**
Well...

Fourier analysis seeks to represent

arbitrary functions by infinite series of

sinusoids. Since

sine is an

even function and

cosine is an

odd function, it's rather handy that we can represent

**any** function by summing a

sine part and a

cosine part!