Editing Function composition (section)

{{short description|Operation on mathematical functions}}
{{about|the mathematical concept|the computer science concept|Function composition (computer science)}}
{{redirect-distinguish|Ring operator|operator ring|operator assistance}}
{{redirects|∘|the character|Degree symbol#Lookalikes}}
{{Use dmy dates|date=May 2019|cs1-dates=y}}
{{Functions}}

In [[mathematics]], the '''composition operator''' <math>\circ</math> takes two [[function (mathematics)|functions]], <math>f</math> and <math>g</math>, and returns a new function <math>h(x) := (g \circ f) (x) = g(f(x))</math>. Thus, the function {{math|''g''}} is [[function application|applied]] after applying {{math|''f''}} to {{math|''x''}}. <math>(g \circ f)</math> is pronounced "the composition of {{math|''g''}} and {{math|''f''}}".<ref>{{Cite web |title=Composition of Functions |url=https://nool.ontariotechu.ca/mathematics/functions/composition-of-functions.php |access-date=2025-02-07 |website=nool.ontariotechu.ca |language=en}}</ref>

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

The composition of functions is a special case of the [[composition of relations]], sometimes also denoted by <math>\circ</math>. As a result, all properties of composition of relations are true of composition of functions,<ref name="Velleman_2006" /> such as [[#Properties|associativity]].