Editing Function composition

{{short description|Operation on mathematical functions}}
{{about|the mathematical concept|the computer science concept|Function composition (computer science)}}
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{{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]].

==Examples==
[[File:Example for a composition of two functions.svg|thumb|Concrete example for the composition of two functions.]]
* Composition of functions on a finite [[set (mathematics)|set]]: If {{math|1=''f'' = {(1, 1), (2, 3), (3, 1), (4, 2)} }}, and {{math|1=''g'' = {(1, 2), (2, 3), (3, 1), (4, 2)} }}, then {{math|1=''g'' ∘ ''f'' = {(1, 2), (2, 1), (3, 2), (4, 3)} }}, as shown in the figure.
* Composition of functions on an [[infinite set]]: If {{math|''f'': '''R''' → '''R'''}} (where {{math|'''R'''}} is the set of all [[real number]]s) is given by {{math|1=''f''(''x'') = 2''x'' + 4}} and {{math|''g'': '''R''' → '''R'''}} is given by {{math|1=''g''(''x'') = ''x''<sup>3</sup>}}, then: {{block indent|text={{math|1=(''f'' ∘ ''g'')(''x'') = ''f''(''g''(''x'')) = ''f''(''x''<sup>3</sup>) = 2''x''<sup>3</sup> + 4}}, and}} {{block indent|text={{math|1=(''g'' ∘ ''f'')(''x'') = ''g''(''f''(''x'')) = ''g''(2''x'' + 4) = (2''x'' + 4)<sup>3</sup>}}.}}
* If an airplane's altitude at time&nbsp;{{mvar|t}} is {{math|''a''(''t'')}}, and the air pressure at altitude {{mvar|x}} is {{math|''p''(''x'')}}, then {{math|(''p'' ∘ ''a'')(''t'')}} is the pressure around the plane at time&nbsp;{{mvar|t}}.
* Function defined on finite sets which change the order of their elements such as [[permutation]]s can be composed on the same set, this being composition of permutations.

==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>

==Composition monoids==
{{main|Transformation monoid}}
Suppose one has two (or more) functions {{math|''f'': ''X'' → ''X'',}} {{math|''g'': ''X'' → ''X''}} having the same domain and codomain; these are often called ''[[Transformation (function)|transformations]]''. Then one can form chains of transformations composed together, such as {{math|''f'' ∘ ''f'' ∘ ''g'' ∘ ''f''}}. Such chains have the [[algebraic structure]] of a [[monoid]], called a ''[[transformation monoid]]'' or (much more seldom) a ''composition monoid''.  In general, transformation monoids can have remarkably complicated structure. One particular notable example is the [[de Rham curve]]. The set of ''all'' functions {{math|''f'': ''X'' → ''X''}} is called the [[full transformation semigroup]]<ref name="Hollings_2014"/> or ''symmetric semigroup''<ref name="Grillet_1995"/> on&nbsp;{{mvar|X}}. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.<ref name="Dömösi-Nehaniv_2005"/>)

[[File:SVG skew and rotation.svg|thumb|Composition of a [[shear mapping]] <small>(red)</small> and a clockwise rotation by 45° <small>(green)</small>. On the left is the original object. Above is shear, then rotate. Below is rotate, then shear.]]
If the given transformations are [[bijective]] (and thus invertible), then the set of all possible combinations of these functions forms a [[transformation group]] (also known as a [[permutation group]]); and one says that the group is [[group generator|generated]] by these functions.

The set of all bijective functions {{math|''f'': ''X'' → ''X''}} (called [[permutation]]s) forms a group with respect to function composition. This is the [[symmetric group]], also sometimes called the ''composition group''. A fundamental result in group theory, [[Cayley's theorem]], essentially says that any group is in fact just a subgroup of a symmetric group ([[up to]] isomorphism).<ref name="Carter_2009"/>

In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called a pseudoinverse) because the symmetric semigroup is a [[regular semigroup]].<ref name="Ganyushkin-Mazorchuk_2008"/>

==Functional powers==
{{main|Iterated function}}
If {{math|''Y'' [[subset|⊆]] ''X''}}, then <math>f:X\to Y</math> may compose with itself; this is sometimes denoted as <math> f^2</math>. That is:
{{block indent|em=1.5|text=<math> (f\circ f)(x) = f(f(x)) = f^2(x)</math>}}
{{block indent|em=1.5|text=<math> (f\circ f \circ f)(x) = f(f(f(x))) = f^3(x)</math>}}
{{block indent|em=1.5|text=<math> (f\circ f\circ f\circ f)(x) = f(f(f(f(x)))) = f^4(x)</math>}}

More generally, for any [[natural number]] {{math|''n'' ≥ 2}}, the {{mvar|n}}th '''functional [[exponentiation|power]]''' can be defined inductively by {{math|1=''f''&thinsp;<sup>''n''</sup> = ''f'' ∘ ''f''&thinsp;<sup>''n''−1</sup> = ''f''&thinsp;<sup>''n''−1</sup> ∘ ''f''}}, a notation introduced by [[Hans Heinrich Bürmann]]{{cn|date=August 2020|reason=The fact is undisputable, but for historical completeness, let's find Bürmann's original work on this and add here as a citation. It must be dated significantly before 1813 (according to Herschel in 1820 und Cajori in 1929.)}}<ref name="Herschel_1820"/><ref name="Cajori_1929"/> and [[John Frederick William Herschel]]<!-- in 1813 -->.<ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Peano_1903"/><ref name="Cajori_1929"/> Repeated composition of such a function with itself is called '''[[iterated function|function iteration]]'''.
* By convention, {{math|''f''&thinsp;<sup>0</sup>}} is defined as the identity map on {{math|''f''&thinsp;}}'s domain, {{math|id<sub>''X''</sub>}}.
* If {{math|1=''Y'' = ''X''}} and {{math|''f'': ''X'' → ''X''}} admits an [[inverse function]] {{math|''f''&thinsp;<sup>−1</sup>}}, negative functional powers {{math|''f''&thinsp;<sup>−''n''</sup>}} are defined for {{math|''n'' > 0}} as the [[additive inverse|negated]] power of the inverse function: {{math|1=''f''&thinsp;<sup>−''n''</sup> = (''f''&thinsp;<sup>−1</sup>)<sup>''n''</sup>}}.<ref name="Herschel_1813"/><ref name="Herschel_1820"/><ref name="Cajori_1929"/>

'''Note:''' If {{mvar|f}} takes its values in a [[ring (mathematics)|ring]] (in particular for real or complex-valued {{math|''f''&thinsp;}}), there is a risk of confusion, as {{math|''f''&thinsp;<sup>''n''</sup>}} could also stand for the {{mvar|n}}-fold product of&nbsp;{{mvar|f}}, e.g. {{math|1=''f''&thinsp;<sup>2</sup>(''x'') = ''f''(''x'') · ''f''(''x'')}}.<ref name="Cajori_1929"/> For trigonometric functions, usually the latter is meant, at least for positive exponents.<ref name="Cajori_1929"/> For example, in [[trigonometry]], this superscript notation represents standard [[exponentiation]] when used with [[trigonometric functions]]:

{{math|1=sin<sup>2</sup>(''x'') = sin(''x'') · sin(''x'')}}.

However, for negative exponents (especially &minus;1), it nevertheless usually refers to the inverse function, e.g., {{math|1=tan<sup>−1</sup> = arctan ≠ 1/tan}}.

In some cases, when, for a given function {{mvar|f}}, the equation {{math|1=''g'' ∘ ''g'' = ''f''}} has a unique solution {{mvar|g}}, that function can be defined as the [[functional square root]] of {{mvar|f}}, then written as {{math|1=''g'' = ''f''&thinsp;<sup>1/2</sup>}}.

More generally, when {{math|1=''g''<sup>''n''</sup> = ''f''}} has a unique solution for some natural number {{math|''n'' > 0}}, then {{math|''f''&thinsp;<sup>''m''/''n''</sup>}} can be defined as {{math|''g''<sup>''m''</sup>}}.

Under additional restrictions,<!---I guess, solvability of g^n = f for all n, and something like [[uniform convergence]] of f^(m/n) for m/n→r, is needed to define f^r for arbitrary r∈'''R'''. A citation is needed about that, anyway.---> this idea can be generalized so that the [[iterated function|iteration count]]  becomes a continuous parameter; in this case, such a system is called a [[flow (mathematics)|flow]], specified through solutions of [[Schröder's equation]]. Iterated functions and flows occur naturally in the study of [[fractals]] and [[dynamical systems]].

To avoid ambiguity, some mathematicians{{cn|date=August 2020|reason=Origin? Example authors?}} choose to use {{math|∘}} to denote the compositional meaning, writing {{math|''f''{{i sup|∘''n''}}(''x'')}} for the {{mvar|n}}-th iterate of the function {{math|''f''(''x'')}}, as in, for example, {{math|''f''{{i sup|∘3}}(''x'')}} meaning {{math|''f''(''f''(''f''(''x'')))}}. For the same purpose, {{math|''f''{{i sup|[''n'']}}(''x'')}} was used by [[Benjamin Peirce]]<ref name="Peirce_1852"/><ref name="Cajori_1929"/> whereas [[Alfred Pringsheim]] and [[Jules Molk]] suggested {{math|{{i sup|''n''}}''f''(''x'')}} instead.<ref name="Pringsheim-Molk_1907"/><ref name="Cajori_1929"/><ref group="nb" name="NB_Rucker"/>

==Alternative notations==
Many mathematicians, particularly in [[group theory]], omit the composition symbol, writing {{math|''gf''}} for {{math|''g'' ∘ ''f''}}.<ref name="Ivanov_2009"/>

During the mid-20th century, some mathematicians adopted [[postfix notation]], writing {{math|''xf''&thinsp;}} for {{math|''f''(''x'')}} and {{math|(''xf'')''g''}} for {{math|''g''(''f''(''x''))}}.<ref name="Gallier_2011" /> This can be more natural than [[prefix notation]] in many cases, such as in [[linear algebra]] when {{mvar|x}} is a [[row vector]] and {{mvar|f}} and {{mvar|g}} denote [[matrix (mathematics)|matrices]] and the composition is by [[matrix multiplication]]. The order is important because function composition is not necessarily commutative. Having successive transformations applying and composing to the right agrees with the left-to-right reading sequence.

Mathematicians who use postfix notation may write "{{math|''fg''}}", meaning first apply {{mvar|f}} and then apply {{mvar|g}}, in keeping with the order the symbols occur in postfix notation, thus making the notation "{{math|''fg''}}" ambiguous. Computer scientists may write "{{math|''f'' ; ''g''}}" for this,<ref name="Barr-Wells_1990"/> thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the [[Z notation]] the ⨾ character is used for left [[relation composition]].<ref name="ISOIEC13568"/> Since all functions are  [[Binary relation#Special types of binary relations|binary relations]], it is correct to use the [fat] semicolon for function composition as well (see the article on [[composition of relations]] for further details on this notation).

==Composition operator==
{{main|Composition operator}}
Given a function&nbsp;{{math|''g''}}, the '''composition operator''' {{math|''C''<sub>''g''</sub>}} is defined as that [[Operator (mathematics)|operator]] which maps functions to functions as <math display="block">C_g f = f \circ g.</math>Composition operators are studied in the field of [[operator theory]].

==In programming languages==
{{main|Function composition (computer science)}}
Function composition appears in one form or another in numerous [[programming language]]s.

==Multivariate functions==
Partial composition is possible for [[multivariate function]]s. The function resulting when some argument {{math|''x''<sub>''i''</sub>}} of the function {{mvar|f}} is replaced by the function {{mvar|g}} is called a composition of {{mvar|f}} and {{mvar|g}} in some computer engineering contexts, and is denoted {{math|1=''f'' {{!}}<sub>''x''<sub>''i''</sub> = ''g''</sub>}}
<math display="block">f|_{x_i = g} = f (x_1, \ldots, x_{i-1}, g(x_1, x_2, \ldots, x_n), x_{i+1}, \ldots, x_n).</math>

When {{mvar|g}} is a simple constant {{mvar|b}}, composition degenerates into a (partial) valuation, whose result is also known as [[Restriction (mathematics)|restriction]] or ''co-factor''.<ref name="Bryant_1986"/>

<math display="block">f|_{x_i = b} = f (x_1, \ldots, x_{i-1}, b, x_{i+1}, \ldots, x_n).</math>

In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of [[primitive recursive function]]. Given {{mvar|f}}, a {{mvar|n}}-ary function, and {{mvar|n}} {{mvar|m}}-ary functions {{math|''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>}}, the composition of {{mvar|f}} with {{math|''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>}}, is the {{mvar|m}}-ary function
<math display="block">h(x_1,\ldots,x_m) = f(g_1(x_1,\ldots,x_m),\ldots,g_n(x_1,\ldots,x_m)).</math>

This is sometimes called the '''generalized composite''' or '''superposition''' of ''f'' with {{math|''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>}}.<ref name="Bergman_2011"/> The partial composition in only one argument mentioned previously can be instantiated from this more general scheme by setting all argument functions except one to be suitably chosen [[projection function]]s. Here {{math|''g''<sub>1</sub>, ..., ''g''<sub>''n''</sub>}} can be seen as a single vector/[[tuple]]-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition.<ref name="Tourlakis_2012"/>

A set of finitary [[operation (mathematics)|operation]]s on some base set ''X'' is called a [[clone (algebra)|clone]] if it contains all projections and is closed under generalized composition. A clone generally contains operations of various [[arity|arities]].<ref name="Bergman_2011"/> The notion of commutation also finds an interesting generalization in the multivariate case; a function ''f'' of arity ''n'' is said to commute with a function ''g'' of arity ''m'' if ''f'' is a [[homomorphism]] preserving ''g'', and vice versa, that is:<ref name="Bergman_2011"/>
<math display="block">f(g(a_{11},\ldots,a_{1m}),\ldots,g(a_{n1},\ldots,a_{nm})) = g(f(a_{11},\ldots,a_{n1}),\ldots,f(a_{1m},\ldots,a_{nm})).</math>

A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity) operation. A binary (or higher arity) operation that commutes with itself is called [[Medial magma|medial or entropic]].<ref name="Bergman_2011"/>

==Generalizations==
[[Composition of relations|Composition]] can be generalized to arbitrary [[binary relation]]s.
If {{math|''R'' ⊆ ''X'' [[cartesian product|×]] ''Y''}} and {{math|''S'' ⊆ ''Y'' × ''Z''}} are two binary relations, then their composition amounts to

<math>R \circ S = \{(x,z) \in X \times Z: (\exists y \in Y)((x,y) \in R\,\and\,(y,z) \in S)\}</math>.

Considering a function as a special case of a binary relation (namely [[Binary relation#Specific_types_of_binary_relations|functional relation]]s), function composition satisfies the definition for relation composition. A small circle {{math|''R''∘''S''}} has been used for the [[Composition_of_relations#Notational_variations|infix notation of composition of relations]], as well as functions. When used to represent composition of functions <math>(g \circ f)(x) \ = \ g(f(x))</math>  however, the text sequence is reversed to illustrate the different operation sequences accordingly.

The composition is defined in the same way for [[partial function]]s and Cayley's theorem has its analogue called the [[Wagner–Preston theorem]].<ref name="Lipcomb_1997"/>

The [[category of sets]] with functions as [[morphism]]s is the prototypical [[Category (mathematics)|category]]. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition.<ref name="Hilton-Wu_1989"/> The structures given by composition are axiomatized and generalized in [[category theory]] with the concept of [[morphism]] as the category-theoretical replacement of functions. The reversed order of composition in the formula {{math|1=(''f'' ∘ ''g'')<sup>−1</sup> = (''g''<sup>−1</sup> ∘ ''f'' <sup>−1</sup>)}} applies for [[composition of relations]] using [[converse relation]]s, and thus in [[group theory]]. These structures form [[dagger category|dagger categories]].<blockquote>''The standard "foundation" for mathematics starts with [[Set theory|sets and their elements]]. It is possible to start differently, by axiomatising not elements of sets but functions between sets. This can be done by using the language of categories and universal constructions.''


''. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories -- specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms'' (''like functions'') ''form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.''

- [[Saunders Mac Lane]], [[Mathematics, Form and Function|Mathematics: Form and Function]]<ref>{{Cite web |title=Saunders Mac Lane - Quotations |url=https://mathshistory.st-andrews.ac.uk/Biographies/MacLane/quotations/ |access-date=2024-02-13 |website=Maths History |language=en}}</ref></blockquote>

==Typography==
The composition symbol {{math|∘}} is encoded as {{unichar|2218|ring operator|html=}}; see the [[Degree symbol#Lookalikes|Degree symbol]] article for similar-appearing Unicode characters. In [[TeX]], it is written <code>\circ</code>.

==See also==
* [[Cobweb plot]] – a graphical technique for functional composition
* [[Combinatory logic]]
* [[Composition ring]], a formal axiomatization of the composition operation
* [[Flow (mathematics)]]
* [[Function composition (computer science)]]
* [[Random variable#Functions of random variables|Function of random variable]], distribution of a function of a random variable
* [[Functional decomposition]]
* [[Functional square root]]
* [[Functional equation]]
* [[Higher-order function]]
* [[Infinite compositions of analytic functions]]
* [[Iterated function]]
* [[Lambda calculus]]

==Notes==
{{Reflist|group="nb"|refs=
<!-- <ref group="nb" name="NB_Dixon_1996">Some authors use {{math|''f'' ∘ ''g'' : ''X'' → ''Z''}}, defined by {{math|1=(''f'' ∘ ''g'' )(''x'') = ''g''(''f''(''x''))}} instead. This is common when a [[postfix notation]] is used, especially if functions are represented by exponents, as, for instance, in the study of [[group action (mathematics)|group action]]s. See {{cite book |author-first1=John D. |author-last1=Dixon |author-first2=Brian |author-last2=Mortimer |title=Permutation groups |date=1996 |publisher=Springer |isbn=0-387-94599-7 |page=[https://archive.org/details/permutationgroup0000dixo/page/5 5] |url=https://archive.org/details/permutationgroup0000dixo/page/5}}</ref> -->
<ref group="nb" name="NB_Strict">The strict sense is used, ''e.g.'', in [[category theory]], where a subset relation is modelled explicitly by an [[inclusion function]].</ref>
<ref group="nb" name="NB_Rucker">[[Alfred Pringsheim]]'s and [[Jules Molk]]'s (1907) notation {{math|{{i sup|''n''}}''f''(''x'')}} to denote function compositions must not be confused with [[Rudolf von Bitter Rucker]]'s (1982) [[Rudy Rucker notation|notation]] {{math|{{i sup|''n''}}''x''}}, introduced by Hans Maurer (1901) and [[Reuben Louis Goodstein]] (1947) for [[tetration]], or with [[David Patterson Ellerman]]'s (1995) {{math|{{i sup|''n''}}''x''}} pre-superscript notation for [[nth root|root]]s.<!-- See {{cite book |title=Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics |chapter=Chapter 12: Parallel Addition, Series-Parallel Duality, and Financial Mathematics: Series Chauvinsism |series=G – Reference, Information and Interdisciplinary Subjects Series |work=The worldly philosophy: studies in intersection of philosophy and economics |author-first=David Patterson |author-last=Ellerman |author-link=David Patterson Ellerman |edition=illustrated |publisher=[[Rowman & Littlefield Publishers, Inc.]] |date=1995-03-21 |isbn=0-8476-7932-2 |pages=237–268 [239] |url=http://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |chapter-url=https://books.google.com/books?id=NgJqXXk7zAAC&pg=PA237 |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20160305012729/http://www.ellerman.org/wp-content/uploads/2012/12/IntellectualTrespassingBook.pdf |archive-date=2016-03-05 |quote=}} [https://web.archive.org/web/20150917191423/http://www.ellerman.org/Davids-Stuff/Maths/sp_math.doc] (271 pages) --><!-- {{cite web |title=Introduction to Series-Parallel Duality |author-first=David Patterson |author-last=Ellerman |author-link=David Patterson Ellerman |publisher=[[University of California at Riverside]] |date=May 2004 |orig-year=1995-03-21 |citeseerx=10.1.1.90.3666 |url=http://www.ellerman.org/wp-content/uploads/2012/12/Series-Parallel-Duality.CV_.pdf |access-date=2019-08-09 |url-status=live |archive-url=https://web.archive.org/web/20190810011716/http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf |archive-date=2019-08-10}} [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.90.3666&rep=rep1&type=pdf] (24 pages) --></ref>
}}

==References==
{{Reflist|refs=
<ref name="Bergman_2011">{{cite book |author-first=Clifford |author-last=Bergman |title=Universal Algebra: Fundamentals and Selected Topics |url=https://books.google.com/books?id=QXi3BZWoMRwC |date=2011 |publisher=[[CRC Press]] |isbn=978-1-4398-5129-6 |pages=[https://books.google.com/books?id=QXi3BZWoMRwC&pg=PA79 79]–80, [https://books.google.com/books?id=QXi3BZWoMRwC&pg=PA90 90]–91}}</ref>
<ref name="Velleman_2006">{{cite book |author-first=Daniel J. |author-last=Velleman |title=How to Prove It: A Structured Approach |url=https://books.google.com/books?id=sXt-ROLLNHcC&pg=PA232 |date=2006 |publisher=[[Cambridge University Press]] |isbn=978-1-139-45097-3 |page=232}}</ref>
<ref name="Rodgers_2000">{{cite book |author-first=Nancy |author-last=Rodgers |title=Learning to Reason: An Introduction to Logic, Sets, and Relations |date=2000 |publisher=[[John Wiley & Sons]] |isbn=978-0-471-37122-9 |pages=359–362 |url=https://books.google.com/books?id=NuN2Iyqzqp4C&q=composition}}</ref>
<ref name="Hollings_2014">{{cite book |author-first=Christopher |author-last=Hollings |title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups |url=https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA334 |date=2014 |publisher=[[American Mathematical Society]] |isbn=978-1-4704-1493-1 |page=334}}</ref>
<ref name="Grillet_1995">{{cite book |author-first=Pierre A. |author-last=Grillet |title=Semigroups: An Introduction to the Structure Theory |url=https://books.google.com/books?id=yM544W1N2UUC&pg=PA2 |date=1995 |publisher=[[CRC Press]] |isbn=978-0-8247-9662-4 |page=2}}</ref>
<ref name="Dömösi-Nehaniv_2005">{{cite book |author-first1=Pál |author-last1=Dömösi |author-first2=Chrystopher L. |author-last2=Nehaniv |title=Algebraic Theory of Automata Networks: An introduction |url=https://books.google.com/books?id=W0i5nfQLOGIC&pg=PA8 |date=2005 |publisher=SIAM |isbn=978-0-89871-569-9 |pages=8}}</ref>
<ref name="Carter_2009">{{cite book |author-first=Nathan |author-last=Carter |title=Visual Group Theory |url=https://books.google.com/books?id=T_o0CnMZecMC&pg=PA95 |date=2009-04-09 |publisher=MAA |isbn=978-0-88385-757-1 |page=95}}</ref>
<ref name="Ganyushkin-Mazorchuk_2008">{{cite book |author-first1=Olexandr |author-last1=Ganyushkin |author-first2=Volodymyr |author-last2=Mazorchuk |title=Classical Finite Transformation Semigroups: An Introduction |url=https://books.google.com/books?id=LC3jxfGEcpYC&pg=PA24 |date=2008 |publisher=[[Springer Science & Business Media]] |isbn=978-1-84800-281-4 |page=24}}</ref>
<ref name="Ivanov_2009">{{cite book |author-first=Oleg A. |author-last=Ivanov |title=Making Mathematics Come to Life: A Guide for Teachers and Students |url=https://books.google.com/books?id=z7EHBAAAQBAJ&pg=PA217 |date=2009-01-01 |publisher=[[American Mathematical Society]] |isbn=978-0-8218-4808-1 |pages=217–}}</ref>
<ref name="Gallier_2011">{{cite book |author-first=Jean |author-last=Gallier |author-link=Jean Gallier |title=Discrete Mathematics |url=https://books.google.com/books?id=HXSjIP0OgCUC&pg=PA118 |date=2011 |publisher=Springer |isbn=978-1-4419-8047-2 |page=118}}</ref>
<ref name="Barr-Wells_1990">{{cite book |author-first1=Michael |author-last1=Barr |author-first2=Charles |author-last2=Wells |title=Category Theory for Computing Science |url=http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf |date=1998 |page=6 |access-date=2014-08-23 |archive-url=https://web.archive.org/web/20160304031956/http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf |archive-date=2016-03-04 |url-status=dead}} (NB. This is the updated and free version of book originally published by [[Prentice Hall]] in 1990 as {{isbn|978-0-13-120486-7}}.)</ref>
<ref name="ISOIEC13568">ISO/IEC 13568:2002(E), p. 23</ref>
<ref name="Bryant_1986">{{cite journal |author-last=Bryant |author-first=R. E. |title=Logic Minimization Algorithms for VLSI Synthesis |journal=IEEE Transactions on Computers |volume=C-35 |issue=8 |date=August 1986 |pages=677–691 |doi=10.1109/tc.1986.1676819 |s2cid=10385726 |url=https://www.cs.cmu.edu/~bryant/pubdir/ieeetc86.pdf}}</ref>
<ref name="Tourlakis_2012">{{cite book |author-first=George |author-last=Tourlakis |title=Theory of Computation |url=https://books.google.com/books?id=zy3M24m5cykC&pg=PA100 |date=2012 |publisher=[[John Wiley & Sons]] |isbn=978-1-118-31533-0 |page=100}}</ref>
<ref name="Lipcomb_1997">{{cite book |author-first=S. |author-last=Lipscomb |title=Symmetric Inverse Semigroups |series=AMS Mathematical Surveys and Monographs |date=1997 |isbn=0-8218-0627-0 |page=xv}}</ref>
<ref name="Hilton-Wu_1989">{{cite book |author-first1=Peter |author-last1=Hilton |author-first2=Yel-Chiang |author-last2=Wu |title=A Course in Modern Algebra |url=https://books.google.com/books?id=ua5gKZt3R6AC&pg=PA65 |date=1989 |publisher=[[John Wiley & Sons]] |isbn=978-0-471-50405-4 |page=65}}</ref>
<ref name="Cajori_1929">{{cite book |author-first=Florian |author-last=Cajori |author-link=Florian Cajori |title=A History of Mathematical Notations |chapter=§472. The power of a logarithm / §473. Iterated logarithms / §533. John Herschel's notation for inverse functions / §535. Persistence of rival notations for inverse functions / §537. Powers of trigonometric functions |volume=2 |orig-year=March 1929 |publisher=[[Open court publishing company]] |location=Chicago, USA |date=1952 |edition=3rd corrected printing of 1929 issue, 2nd |pages=108, 176–179, 336, 346 |isbn=978-1-60206-714-1 |url=https://books.google.com/books?id=bT5suOONXlgC |access-date=2016-01-18 |quote=[…] §473. ''Iterated logarithms'' […] We note here the symbolism used by [[Alfred Pringsheim|Pringsheim]] and [[Jules Molk|Molk]] in their joint ''Encyclopédie'' article: "<sup>2</sup>log<sub>''b''</sub>&thinsp;''a'' = log<sub>''b''</sub> (log<sub>''b''</sub>&thinsp;''a''), …, <sup>''k''+1</sup>log<sub>''b''</sub>&thinsp;''a'' = log<sub>''b''</sub> (<sup>''k''</sup>log<sub>''b''</sub>&thinsp;''a'')."{{citeref|Pringsheim|Molk|1907|a<!-- [10] -->}} […] §533. ''[[John Frederick William Herschel|John Herschel]]'s notation for inverse functions,'' sin<sup>&minus;1</sup>&thinsp;''x'', tan<sup>&minus;1</sup>&thinsp;''x'', etc., was published by him in the ''[[Philosophical Transactions of London]]'', for the year 1813. He says ({{citeref|Herschel|1813|p.&nbsp;10|style=plain}}): "This notation cos.<sup>&minus;1</sup>&thinsp;''e'' must not be understood to signify 1/cos.&nbsp;''e'', but what is usually written thus, arc&thinsp;(cos.=''e'')." He admits that some authors use cos.<sup>''m''</sup>&thinsp;''A'' for (cos.&thinsp;''A'')<sup>''m''</sup>, but he justifies his own notation by pointing out that since ''d''<sup>2</sup>&thinsp;''x'', Δ<sup>3</sup>&thinsp;''x'', Σ<sup>2</sup>&thinsp;''x'' mean ''dd''&thinsp;''x'', ΔΔΔ&thinsp;''x'', ΣΣ&thinsp;''x'', we ought to write sin.<sup>2</sup>&thinsp;''x'' for sin.&thinsp;sin.&thinsp;''x'', log.<sup>3</sup>&thinsp;''x'' for log.&thinsp;log.&thinsp;log.&thinsp;''x''. Just as we write ''d''<sup>&minus;''n''</sup>&thinsp;V=∫<sup>''n''</sup>&thinsp;V, we may write similarly sin.<sup>&minus;1</sup>&thinsp;''x''=arc&thinsp;(sin.=''x''), log.<sup>&minus;1</sup>&thinsp;''x''.=c<sup>''x''</sup>. Some years later Herschel explained that in 1813 he used ''f''<sup>''n''</sup>(''x''), ''f''<sup>&minus;''n''</sup>(''x''), sin.<sup>&minus;1</sup>&thinsp;''x'', etc., "as he then supposed for the first time. The work of a German Analyst, [[Hans Heinrich Bürmann|Burmann]], has, however, within these few months come to his knowledge, in which the same is explained at a considerably earlier date. He[Burmann], however, does not seem to have noticed the convenience of applying this idea to the inverse functions tan<sup>&minus;1</sup>, etc., nor does he appear at all aware of the inverse calculus of functions to which it gives rise." Herschel adds, "The symmetry of this notation and above all the new and most extensive views it opens of the nature of analytical operations seem to authorize its universal adoption."{{citeref|Herschel|1820|b<!-- [4] -->}} […] §535. ''Persistence of rival notations for inverse function.''&mdash; […] The use of Herschel's notation underwent a slight change in [[Benjamin Peirce]]'s books, to remove the chief objection to them; Peirce wrote: "cos<sup>[&minus;1]</sup>&thinsp;''x''," "log<sup>[&minus;1]</sup>&thinsp;''x''."{{citeref|Peirce|1852|c<!-- [1] -->}} […] §537. ''Powers of trigonometric functions.''&mdash;Three principal notations have been used to denote, say, the square of sin&thinsp;''x'', namely, (sin&thinsp;''x'')<sup>2</sup>, sin&thinsp;''x''<sup>2</sup>, sin<sup>2</sup>&thinsp;''x''. The prevailing notation at present is sin<sup>2</sup>&thinsp;''x'', though the first is least likely to be misinterpreted. In the case of sin<sup>2</sup>&thinsp;''x'' two interpretations suggest themselves; first, sin&thinsp;''x'' &sdot; sin&thinsp;''x''; second,{{citeref|Peano|1903|d<!-- [8] -->}} sin&thinsp;(sin&thinsp;''x''). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log<sup>2</sup>&thinsp;''x'', where log&thinsp;''x'' &sdot; log&thinsp;''x'' and log&thinsp;(log&thinsp;''x'') are of frequent occurrence in analysis. […] The notation sin<sup>''n''</sup>&thinsp;''x'' for (sin&thinsp;''x'')<sup>''n''</sup> has been widely used and is now the prevailing one. […]}} (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)</ref>
<ref name="Herschel_1813">{{cite journal |author-first=John Frederick William |author-last=Herschel |author-link=John Frederick William Herschel |title=On a Remarkable Application of Cotes's Theorem |journal=[[Philosophical Transactions of the Royal Society of London]] |publisher=[[Royal Society of London]], printed by W. Bulmer and Co., Cleveland-Row, St. James's, sold by G. and W. Nicol, Pall-Mall |location=London |volume=103 |number=Part 1 |date=1813 |orig-year=1812-11-12 |jstor=107384 |pages=8–26 [10]|doi=10.1098/rstl.1813.0005 |s2cid=118124706 |doi-access= }}</ref>
<ref name="Herschel_1820">{{cite book |author-first=John Frederick William |author-last=Herschel |author-link=John Frederick William Herschel |title=A Collection of Examples of the Applications of the Calculus of Finite Differences |chapter=Part III. Section I. Examples of the Direct Method of Differences |location=Cambridge, UK |publisher=Printed by J. Smith, sold by J. Deighton & sons |date=1820 |pages=1–13 [5–6] |chapter-url=https://books.google.com/books?id=PWcSAAAAIAAJ&pg=PA5 |access-date=2020-08-04 |url-status=live |archive-url=https://web.archive.org/web/20200804031020/https://books.google.de/books?hl=de&id=PWcSAAAAIAAJ&jtp=5 |archive-date=2020-08-04}} [https://archive.org/details/acollectionexam00lacrgoog] (NB. Inhere, Herschel refers to his {{citeref|Herschel|1813|1813 work|style=plain}} and mentions [[Hans Heinrich Bürmann]]'s older work.)</ref>
<ref name="Peirce_1852">{{cite book |author-first=Benjamin |author-last=Peirce |author-link=Benjamin Peirce |title=Curves, Functions and Forces |volume=I |edition=new |location=Boston, USA |date=1852 |page=203}}</ref>
<ref name="Peano_1903">{{cite book |author-first=Giuseppe |author-last=Peano |author-link=Giuseppe Peano |title=Formulaire mathématique |language=fr |volume=IV |date=1903 |page=229}}</ref>
<ref name="Pringsheim-Molk_1907">{{cite book |author-first1=Alfred |author-last1=Pringsheim |author-link1=Alfred Pringsheim |author-first2=Jules |author-last2=Molk |author-link2=Jules Molk |title=Encyclopédie des sciences mathématiques pures et appliquées |language=fr |id=Part I |volume=I |date=1907 |page=195}}</ref>
}}

==External links==
* {{springer|title=Composite function|id=p/c024260}}
* "[http://demonstrations.wolfram.com/CompositionOfFunctions/ Composition of Functions]" by Bruce Atwood, the [[Wolfram Demonstrations Project]], 2007.

[[Category:Functions and mappings]]
[[Category:Basic concepts in set theory]]
[[Category:Binary operations]]