Function composition

In mathematics, the composition operator ${\displaystyle \circ }$ takes two functions, ${\displaystyle f}$ and ${\displaystyle g}$, and returns a new function ${\displaystyle h(x):=(g\circ f)(x)=g(f(x))}$. Thus, the function g is applied after applying f to x. ${\displaystyle (g\circ f)}$ is pronounced "the composition of g and f".^[1]

Reverse composition, sometimes denoted ${\displaystyle f\mapsto g}$ , applies the operation in the opposite order, applying ${\displaystyle f}$ first and ${\displaystyle g}$ second. Intuitively, reverse composition is a chaining process in which the output of function f feeds the input of function g.

The composition of functions is a special case of the composition of relations, sometimes also denoted by ${\displaystyle \circ }$. As a result, all properties of composition of relations are true of composition of functions,^[2] such as associativity.