A function describes a rule that assigns each allowed input exactly one output. If the function is called $f$, the output corresponding to an input $x$ is written

$$f(x).$$

For example, if

$$f(x)=2x+3,$$

then $f(4)=11$.

Domain, codomain and image

A function written

$$f:A\to B$$

has domain $A$, the set of allowed inputs, and codomain $B$, the set in which outputs are declared to lie. The outputs actually produced form the image of the function.

A formula may impose restrictions on its domain. For example,

$$f(x)=\frac1{x-2}$$

is not defined at $x=2$.

Functions and graphs

The graph of a real-valued function contains the points

$$(x,f(x)).$$

Because one input cannot have two different outputs, a vertical line can intersect the graph of a function at most once.

Composition

Functions can be applied one after another. If $g$ produces an output that can be used as an input to $f$, their composition is

$$(f\circ g)(x)=f(g(x)).$$

For example, if $g(x)=x^2$ and $f(x)=2x+3$, then

$$(f\circ g)(x)=2x^2+3.$$

Inverse functions

Some functions can be reversed. If $f$ is one-to-one on the domain being considered, its inverse $f^{-1}$ satisfies

$$f^{-1}(f(x))=x.$$

The inverse exchanges the roles of input and output; it is not the reciprocal $1/f(x)$.