Editing Function composition (section)

==Properties==
The composition of functions is always [[associative]]—a property inherited from the [[composition of relations]].<ref name="Velleman_2006"/> That is, if {{mvar|f}}, {{mvar|g}}, and {{mvar|h}} are composable, then {{math|1=''f'' ∘ (''g'' ∘ ''h'') = (''f'' ∘ ''g'') ∘ ''h''}}.<ref name=":0">{{Cite web|last=Weisstein|first=Eric W.|title=Composition|url=https://mathworld.wolfram.com/Composition.html|access-date=2020-08-28|website=mathworld.wolfram.com|language=en}}</ref> Since the parentheses do not change the result, they are generally omitted.

In a strict sense, the composition {{math|1=''g'' ∘ ''f''}} is only meaningful if the codomain of {{mvar|f}} equals the domain of {{mvar|g}}; in a wider sense, it is sufficient that the former be an improper [[subset]] of the latter.<ref group="nb" name="NB_Strict"/>
Moreover, it is often convenient to tacitly restrict the domain of {{mvar|f}}, such that {{mvar|f}} produces only values in the domain of {{mvar|g}}. For example, the composition {{math|1=''g'' ∘ ''f''}} of the functions {{math|''f'' : [[real number|'''R''']] → [[interval (mathematics)#Infinite endpoints|(−∞,+9] ]]}} defined by {{math|1=''f''(''x'') = 9 − ''x''<sup>2</sup>}} and {{math|''g'' : [[interval (mathematics)#Infinite endpoints|[0,+∞)]] → '''R'''}} defined by <math>g(x) = \sqrt x</math> can be defined on the [[interval (mathematics)|interval]] {{math|[−3,+3]}}.

[[Image:Absolute value composition.svg|thumb|upright=1|Compositions of two [[Real number|real]] functions, the [[absolute value]] and a [[cubic function]], in different orders, show a non-commutativity of composition.]]
The functions {{mvar|g}} and {{mvar|f}} are said to [[commutative|commute]] with each other if {{math|1=''g'' ∘ ''f'' = ''f'' ∘ ''g''}}. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, {{math|1={{abs|''x''}} + 3 = {{abs|''x'' + 3}}}} only when {{math|''x'' ≥ 0}}. The picture shows another example.

The composition of [[One-to-one function|one-to-one]] (injective) functions is always one-to-one. Similarly, the composition of [[Onto function|onto]] (surjective) functions is always onto. It follows that the composition of two [[bijection]]s is also a bijection. The [[inverse function]] of a composition (assumed invertible) has the property that {{math|1=(''f'' ∘ ''g'')<sup>−1</sup> = ''g''<sup>−1</sup>∘ ''f''<sup>−1</sup>}}.<ref name="Rodgers_2000"/>

[[Derivative]]s of compositions involving differentiable functions can be found using the [[chain rule]]. [[Higher derivative]]s of such functions are given by [[Faà di Bruno's formula]].<ref name=":0" />

Composition of functions is sometimes described as a kind of [[multiplication]] on a function space, but has very different properties from [[pointwise]] multiplication of functions (e.g. composition is not [[Commutative property|commutative]]).<ref>{{Cite web |date=2020-01-16 |title=3.4: Composition of Functions |url=https://math.libretexts.org/Courses/Western_Connecticut_State_University/Draft_Custom_Version_MAT_131_College_Algebra/03%3A_Functions/3.04%3A_Composition_of_Functions |access-date=2020-08-28 |website=Mathematics LibreTexts |language=en}}</ref>