Every Function Is The Sum of an Even and Odd Function
Theorem
Every function \(f : \mathbb{R} \to \mathbb{R}\) can be written as the sum of an even and an odd function.
Proof
Suppose \(f : \mathbb{R} \to \mathbb{R}\) is a function, and define
\[ f_\text{even}(x) = \frac{f(x) + f(-x)}{2} \quad \text{and} \quad f_\text{odd}(x) = \frac{f(x) - f(-x)}{2}.\]
As such,
\[ f_\text{even}(x) + f_\text{odd}(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2} = f(x)\]
while we have that
\[ f_\text{even}(-x) = \frac{f(-x) + f(x)}{2} = \frac{f(x) + f(-x)}{2} = f_\text{even}(x)\]
and
\[ -f_\text{odd}(-x) = \frac{-f(-x) + f(x)}{2} = \frac{f(x) - f(-x)}{2} = f_\text{odd}(x).\]
Thus \(f_\text{odd} + f_\text{even} = f\) is the decomposition of \(f\) as the sum of an even and odd function.