# A Simple Oddity Trick

A function is

Even: f(-x) = f(x)
Odd: f(-x) = -f(x)

So

f(x)
= [f(x)+f(-x)]/2 + [f(x)-f(-x)]/2
= f_even + f_odd


Ex.
If f(x) = exp(x),
f_even = [exp(x)+exp(-x)]/2 = cosh(x),
f_odd = [exp(x)-exp(-x)]/2 = sinh(x).

exp(x) = cosh(x) + sinh(x).


Ex.
If f(x) = x^3 + 3x^2 - 4x + 7,
f_even = 3x^2 + 7,
f_odd = x^3 - 4x.

x^3 + 3x^2 - 4x + 7 = (3x^2 + 7) + (x^3 - 4x).

and f(a)+f(-a) = 2 * (3a^2 + 7) = 6a^2 + 14.