Even & Odd Functions

Symmetry simplifies analysis. Even and odd functions have specific symmetry properties that make them easier to graph and study.

Definitions

📐Definition — Even Function

A function $f$ is even if for every $x$ in its domain:

$f(-x) = f(x)$

The graph of an even function is symmetric about the $y$ -axis.

📐Definition — Odd Function

A function $f$ is odd if for every $x$ in its domain:

$f(-x) = -f(x)$

The graph of an odd function is symmetric about the origin (180° rotational symmetry).

Important prerequisite: The domain must be symmetric about $0$ . That is, if $x$ is in the domain, then $-x$ must be as well. For example, $f(x) = \sqrt{x}$ (domain $[0, +\infty)$ ) cannot be even or odd.

How to Test

To determine whether $f$ is even, odd, or neither:

Compute $f(-x)$ by substituting $-x$ for every $x$ .
Simplify the result.
Compare: if $f(-x) = f(x)$ , the function is even. If $f(-x) = -f(x)$ , it is odd. Otherwise, neither.

Examples

Function	$f(-x)$	Classification
$x^2$	$(-x)^2 = x^2 = f(x)$	Even
$x^4 + 2x^2$	$x^4 + 2x^2 = f(x)$	Even
$x^3$	$(-x)^3 = -x^3 = -f(x)$	Odd
$x^3 - x$	$-x^3 + x = -(x^3 - x) = -f(x)$	Odd
$x^2 + x$	$x^2 - x \neq f(x)$ and $\neq -f(x)$	Neither

From trigonometry: $\cos x$ is even, $\sin x$ is odd.

Worked Example

✎Example — Testing for Parity

Determine whether $f(x) = x^4 - 3x^2 + 1$ is even, odd, or neither.

Solution. Compute $f(-x)$ :

$f(-x) = (-x)^4 - 3(-x)^2 + 1 = x^4 - 3x^2 + 1 = f(x)$

Since $f(-x) = f(x)$ , the function is even.

Decomposition into Even and Odd Parts

An interesting fact: any function $f$ (with a symmetric domain) can be written as the sum of an even function and an odd function:

$f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}_{\text{even part}} + \underbrace{\frac{f(x) - f(-x)}{2}}_{\text{odd part}}$

You can verify that the first term satisfies $g(-x) = g(x)$ and the second satisfies $h(-x) = -h(x)$ .

✏Exercise

Classify $g(x) = x^3 + x^2$ . Is it even, odd, or neither? Compute $g(-x)$ and compare.