Function composition

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the composition operator 
 
 
 
 ∘
 
 
 {\displaystyle \circ }
 
 takes two <a href="/facts/Function_(mathematics)/CdReEKCh">functions</a>, 
 
 
 
 f
 
 
 {\displaystyle f}
 
 and 
 
 
 
 g
 
 
 {\displaystyle g}
 
, and returns a new function 
 
 
 
 h
 (
 x
 )
 :=
 (
 g
 ∘
 f
 )
 (
 x
 )
 =
 g
 (
 f
 (
 x
 )
 )
 
 
 {\displaystyle h(x):=(g\circ f)(x)=g(f(x))}
 
. Thus, the function g is <a href="/facts/Function_application/p4auE0QF">applied</a> after applying f to x. 
 
 
 
 (
 g
 ∘
 f
 )
 
 
 {\displaystyle (g\circ f)}
 
 is pronounced "the composition of g and f".
Reverse composition, sometimes denoted 
 
 
 
 f
 ↦
 g
 
 
 {\displaystyle f\mapsto g}
 
 , applies the operation in the opposite order, applying 
 
 
 
 f
 
 
 {\displaystyle f}
 
 first and 
 
 
 
 g
 
 
 {\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 <a href="/facts/Composition_of_relations/tD8joBlo">composition of relations</a>, sometimes also denoted by 
 
 
 
 ∘
 
 
 {\displaystyle \circ }
 
. As a result, all properties of composition of relations are true of composition of functions, such as associativity.

Function composition open-in-new

Function composition