📂Functions

Any Function Can Always Be Expressed as the Sum of Odd and Even Functions

Theorem

The arbitrary function $f$ defined in $\mathbb{R}$ can always be expressed as a sum of an even function and an odd function.

Proof

Let $f_{e}(t)$ and $f_o(t)$ be as follows.

$$f_{e}(t)=\dfrac{ f(t)+f(-t)}{2},\ \ \ f_o(t)=\dfrac{ f(t)-f(-t)}{2}$$

Then, $f_{e}(t)$ is an even function, and $f_o(t)$ is an odd function, and the following equation holds.

$$f_{e}(x)+f_o(x)=f(x)$$

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