Even and odd functions

Definition

Let $f \colon \mathbb{R} \to \mathbb{R}$ be a function. The function $f$ is even if $f(-x) = f(x)$ for all $x \in \mathbb{R}$. It is odd if $f(-x) = -f(x)$ for all $x \in \mathbb{R}$.

Remarks

The graph of an even function is symmetric about the $y$-axis. The graph of an odd function is symmetric about the origin. For example, $x^2$ and $\cos x$ are even, and $x^3$ and $\sin x$ are odd. Each function splits in a unique way into an even part and an odd part: $f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$. Leonhard Euler introduced the two notions in 1727. Mathlib defines even functions as Function.Even.

Explanations

L1 human

An even function takes the same value at $x$ and at $-x$. An odd function takes opposite values at $x$ and at $-x$.