Composition of functions

A composition of functions is the function obtained by applying one function after another: if $f:A\to B$ and $g:B\to C$ are functions (so the codomain of $f$ matches the domain of $g$ ), then the composition $g\circ f:A\to C$ is defined by

(g\circ f)(a)=g(f(a))\quad\text{for all }a\in A.

Remarks

Composition is associative when defined, and identity functions act as identities for composition. If $f$ is bijective, then composing with its inverse function recovers the appropriate identity functions.

Examples

Let $f:\mathbb{Z}\to\mathbb{Z}$ be $f(n)=n+1$ and let $g:\mathbb{Z}\to\mathbb{Z}$ be $g(n)=2n$ ; then $(g\circ f)(n)=2(n+1)=2n+2$ .
If $i:S\to A$ is the inclusion of a subset and $f:A\to B$ is any function, then $f\circ i:S\to B$ is the restriction of $f$ to $S$ .