# Every function can be written as a sum of an even and an odd function

Let $f(x)$ be any well-defined function.

We want to express it as a sum of an even function and an odd function.

Let us define two other functions as follows:

$F(x) = \frac{f(x)+f(-x)}{2}$ and $G(x)=\frac{f(x)-f(-x)}{2}$.

Claim I: F(x) is an even function.

Proof I; Since by definition $F(x)= \frac{f(x)+f(-x)}{2}$, so $F(-x) = \frac{f(-x) +f(-(-x))}{2}=\frac{f(-x)+f(x)}{2} \Longrightarrow F(x) = F(-x)$ so that F(x) is indeed an even function.

Claim 2: G(x) is an odd function.

Proof 2: Since by definition $G(x) = \frac{f(x)-f(-x)}{2}$, so $G(-x) = \frac{f(-x)-f(-(-x))}{2} = \frac{f(-x)-f(x)}{2} = -\frac{f(x)-f(-x)}{2} = -G(-x) \Longrightarrow G(x) = -G(-x)$ so that G(x) is indeed an odd function.

Claim 3: $f(x)= F(x) + G(x)$

Proof 3: $F(x)+ G(x) = \frac{f(x)+f(-x)}{2} + \frac{f(x)-f(-x)}{2} = \frac{f(x)}{2} + \frac{f(-x)}{2} + \frac{f(x)}{2} - \frac{f(-x)}{2} = f(x)$ indeed.

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