# Definition: Composition (of functions)

Definition: Let $f:A\rightarrow B$ and $g:B\rightarrow C$ be two functions, with their domains and codomains as written.  The composition of $f$ and $g$, denoted by $g\circ f$, is the function $g\circ f: A\rightarrow C$ that takes $a\in A$ to $g(f(a))\in C$—this last phrase is often denoted by $a\mapsto g(f(a))$.  In other words, $g\circ f(a)=g(f(a))$.

This is an extremely important definition, and for more details check out lesson 31!