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) = fe(x) +  fo(x)
       even   odd

then
 f(-x) = fe(-x) +  fo(-x) = fe(x) - fo(x)

Solving continuously gives:

 fe(x) = 1/2 (f(x) + f(-x))
and
 fo(t) = 1/2 (f(x) - f(-x))  !

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*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!

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