Even and Odd Functions

A separate and very useful way of classifying a real function looks at the symmetry of its rule — how $f(-x)$ compares with $f(x)$. This classification only makes sense when the domain is symmetric about the origin (i.e. whenever $x$ is in the domain, so is $-x$).

Even function. A function $f$ is called even if

Replacing $x$ by $-x$ leaves the output unchanged. The graph of an even function is symmetric about the $y$-axis — the left half is the mirror image of the right half. Standard examples are $f(x)=x^2$, $f(x)=x^4$, $f(x)=|x|$ and $f(x)=\cos x$.

Odd function. A function $f$ is called odd if

Replacing $x$ by $-x$ reverses the sign of the output. The graph of an odd function has rotational symmetry about the origin — rotating it through $180^\circ$ about the origin leaves it unchanged. Standard examples are $f(x)=x$, $f(x)=x^3$ and $f(x)=\sin x$. Note that for any odd function whose domain contains $0$, we must have $f(0)=-f(0)$, hence $f(0)=0$ — the graph passes through the origin.

Neither. Most functions are neither even nor odd. For example $f(x)=x^2+x$ gives $f(-x)=x^2-x$, which equals neither $f(x)$ nor $-f(x)$, so it is neither even nor odd.

The working test is purely algebraic: compute $f(-x)$, simplify, and compare it with $f(x)$ and with $-f(x)$.

• If $f(-x)=f(x)$ throughout — even. …