Even and Odd Functions

Type	Condition	Graph Symmetry
Even	$f(-x)=f(x)$	Symmetric about the $y$-axis
Odd	$f(-x)=-f(x)$	Symmetric about the origin

Even Functions

A function $y=f(x)$ defined on an interval $(-a,a)$ is called even if

$$f(-x)=f(x)\text{for every }x\text{ in }(-a,a).$$

If a function is even, its graph is symmetric about the $y$-axis. Geometrically: if you fold the graph along the $y$-axis, the two halves coincide exactly.

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

Examples of even functions: $f(x)=x^2$, $f(x)=x^4$, $f(x)=\cos x$, $f(x)=|x|$.

Odd Functions

A function $y=f(x)$ defined on an interval $(-a,a)$ is called odd if

$$f(-x)=-f(x)\text{for every }x\text{ in }(-a,a).$$

If a function is odd, its graph is symmetric about the origin. Geometrically: if you rotate the graph by $180°$ about the origin, it looks identical. Equivalently, for any point $(x,y)$ on the graph, the point $(-x,-y)$ is also on the graph.

Graph of an odd function is symmetric about the origin.

Examples of odd functions: $f(x)=x$, $f(x)=x^3$, $f(x)=\sin x$, $f(x)=\tan x$.

Frequently Asked Questions

Must a function be either even or odd?

No. Most functions are neither even nor odd. For example, $f(x)=x+1$ is neither: $f(-x)=-x+1\ne x+1=f(x)$ (not even) and $f(-x)=-x+1\ne -(x+1)=-f(x)$ (not odd). The zero function $f(x)=0$ is the only function that is both even and odd.

How can I check whether a function is even or odd?

Substitute $-x$ for $x$ in the formula and simplify. If the result equals the original formula, the function is even. If the result equals the negative of the original formula, the function is odd. If neither, the function is neither even nor odd.

Why does the integral of an odd function over $[-a,a]$ equal zero?

By symmetry: for every positive contribution $f(x)dx$ on the right half $[0,a]$, there is an equal negative contribution $f(-x)dx=-f(x)dx$ on the left half $[-a,0]$. They cancel exactly. This is a very useful shortcut: ${\int }_{-a}^a\sin xdx=0$ is obvious without computing anything.