|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| {{For|function composition in computer science|function composition (computer science)}}
| | Nothing to say about myself really.<br>Yes! Im a member of wmflabs.org.<br>I really hope I am useful in some way .<br><br>My weblog :: [http://hemorrhoidtreatmentfix.com/internal-hemorrhoids internal hemorrhoid] |
| [[Image:Compfun.svg|thumb|upright=1.5|{{math|''g'' ∘ ''f'' }}, the '''composition''' of {{math|''f''}} and {{math|''g''}}. For example, {{math|1=(''g'' ∘ ''f'' )(c) = #}}.]]
| |
| | |
| In [[mathematics]], '''function composition''' is the [[pointwise]] application of one [[function (mathematics)|function]] to another to produce a third function. For instance, the functions {{math|''f'' : ''X'' → ''Y''}} and {{math|''g'' : ''Y'' → ''Z''}} can be ''composed'' to yield a function which maps {{mvar|x}} in {{mvar|X}} to {{mvar|g(f(x))}} in {{mvar|Z}}. Intuitively, if {{mvar|z}} is a function {{mvar|g}} of {{mvar|y}} and {{mvar|y}} is a function {{mvar|f}} of {{mvar|x}}, then {{mvar|z}} is a function of {{mvar|x}}.
| |
| | |
| The resulting ''composite'' function, notated {{math|''g'' ∘ ''f'' : ''X'' → ''Z''}} --- interchangeable written, in many sources and within this article, as {{math|''g'' ∘ ''f'' : ''X'' → ''Y''}} --- is defined by {{math|1=(''g'' ∘ ''f'' )(''x'') = ''g''(''f''(''x''))}} for all {{math|''x''}} in {{math|''X''}}. The notation {{math|''g'' ∘ ''f''}} is read as "{{math|''g''}} circle {{math|''f''}}", or "{{math|''g''}} round {{math|''f''}}", or "{{math|''g''}} composed with {{math|''f''}}", "{{math|''g''}} after {{math|''f''}}", "{{math|''g''}} following {{math|''f''}}", or "{{math|''g''}} of {{math|''f''}}".
| |
| <br clear="right"/> | |
| | |
| [[Image:Absolute value composition.svg|thumb|upright=1|Compositions of two [[real number|real]] functions, [[absolute value]] and a [[cubic function]], in different orders show a non-commutativity of the composition.]]
| |
| | |
| {{Functions}}
| |
| | |
| The composition of functions is always [[associative]]. That is, if {{math|''f''}}, {{math|''g''}}, and {{math|''h''}} are three functions with suitably chosen [[Domain of a function|domain]]s and [[codomain]]s, then {{math|1=''f'' ∘ (''g'' ∘ ''h'') = (''f'' ∘ ''g'') ∘ ''h''}}, where the parentheses serve to indicate that composition is to be performed first for the parenthesized functions. Since there is no distinction between the choices of placement of parentheses, they may be safely left off.
| |
| | |
| <!-- If somebody is willing to move this out of the lead, then do not forget to withdraw Image:Absolute value composition.svg (above) as well, please -->The functions {{mvar|g}} and {{mvar|f}} are said to [[commutative|commute]] with each other if {{math|1=''g'' ∘ ''f'' = ''f'' ∘ ''g''}}. In general, composition of functions will not be commutative. 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}}.
| |
| | |
| [[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]].
| |
| | |
| ==Example==
| |
| [[File:Academ Example of similarity with ratio square root of 2.svg|thumb|upright=1.2|The [[Similarity (geometry)|similarity]] that transforms triangle ''EFA'' into triangle ''ATB'' is the composition of a [[Homothetic transformation|homothety]] {{math|''H''}} and a [[Rotation (mathematics)|rotation]] {{math|''R''}}, of which the common centre is ''S. '' For example, [[Image (mathematics)|the image]] of ''A '' under the rotation {{math|''R''}} is ''U'', which may be written {{nowrap|1={{math|''R ''}}(''A'') = ''U. ''}} {{nowrap|1=And {{math|''H''}}(''U'') = ''B'' }} means that the [[map (mathematics)|mapping]] {{math|''H''}} transforms {{Nowrap|''U'' into ''B. ''}} {{nowrap|1=Thus {{math|''H''(''R'' }}(''A'')) = {{math|(''H ∘ R '')}}(''A'') = ''B''}}.]]
| |
| As an example, suppose that an airplane's elevation at time {{mvar|t}} is given by the function {{math|''h''(''t'')}} and that the oxygen concentration at elevation {{mvar|x}} is given by the function {{math|''c''(''x'')}}.
| |
| Then {{math|(''c'' ∘ ''h'')(''t'')}} describes the oxygen concentration around the plane at time {{mvar|t}}.
| |
| | |
| ==Functional powers==
| |
| {{main|Iterated function}}
| |
| If [[subset|<math>Y \subseteq X</math>]] then <math>f\colon X\rightarrow Y</math> may compose with itself; this is sometimes denoted as ''f'' <sup>2</sup>. Thus:
| |
| | |
| :<math>(f\circ f)(x) = f(f(x)) = f^2(x)</math>
| |
| | |
| :<math>(f\circ f\circ f)(x) = f(f(f(x))) = f^3(x)</math>
| |
| | |
| Repeated composition of a function with itself is called '''[[iterated function]]'''.
| |
| | |
| The '''functional [[Exponentiation|powers]]'''
| |
| <math>f\circ f^n=f^n\circ f=f^{n+1}</math>
| |
| for [[natural number|natural]] ''n'' follow immediately.
| |
| * By convention, <math>f^0= id_{D(f)}</math> <math>\big(</math>the identity map on the domain of ''f'' ).
| |
| * If <math>f\colon X\rightarrow X</math> admits an [[inverse function]], negative functional powers <math>f^{-k}\,</math> <math>(k>0)</math> are defined as the [[additive inverse|opposite]] power of the inverse function, <math>(f^{-1})^k</math>.
| |
| | |
| '''Note:''' If {{mvar|f}} takes its values in a [[ring (mathematics)|ring]] (in particular for real or complex-valued {{mvar|f}}), there is a risk of confusion, as {{math|''f''<sup> ''n''</sup>}} could also stand for the {{mvar|n}}-fold product of {{mvar|f}}, e.g. {{math|1=''f''<sup> 2</sup>(''x'') = ''f''(''x'') · ''f''(''x'')}}. For trigonometric functions, usually the latter is meant, at least for positive exponents. For example, in [[trigonometric identity|trigonometry]], this superscript notation represents standard [[exponentiation]] when used with [[trigonometric function]]s:
| |
| {{math|1=sin<sup>2</sup>(''x'') = sin(''x'') · sin(''x'')}}.
| |
| However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., {{math|1=tan<sup>−1</sup> = arctan (≠ 1/tan)}}.
| |
| | |
| In some cases, an expression for {{mvar|f}} in {{math|''g''(''x'') {{=}} ''f'' <sup>''r''</sup>(''x'')}} can be derived from the rule for {{mvar|g}} given non-integer values of {{mvar|r}}. This is called [[fractional iteration]]. For instance, a [[functional square root|half iterate]] of a function {{mvar|f}} is a function {{mvar|g}} satisfying {{math|''g''(''g''(''x'')) {{=}} ''f''(''x'').}} Another example would be that where {{mvar|f}} is the [[successor function]], {{math|''f''<sup> ''r''</sup>(''x'') {{=}} ''x'' + ''r''}}. 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]].
| |
| | |
| ==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. Then one can form long, potentially complicated chains of these functions composed together, such as {{math|''f'' ∘ ''f'' ∘ ''g'' ∘ ''f''}}. Such long chains have the [[algebraic structure]] of a [[monoid]], called [[transformation monoid]] or [[composition monoid]]. In general, composition 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]] on {{mvar|X}}.
| |
| | |
| If the functions are [[bijective]], then the set of all possible combinations of these functions forms a [[transformation group]]; and one says that the group is [[group generator|generated]] by these functions.
| |
| | |
| The set of all [[bijective]] functions {{math|''f'': ''X'' → ''X''}} forms a group with respect to the composition operator. This is the [[symmetric group]], also sometimes called the '''composition group'''.
| |
| | |
| ==Alternative notations==
| |
| | |
| {{Unreferenced section|date=May 2013}}
| |
| | |
| *Many mathematicians omit the composition symbol, writing {{math|''gf''}} for {{math|''g'' ∘ ''f''}}.
| |
| | |
| *In the mid-20th century, some mathematicians decided that writing "{{math|''g'' ∘ ''f'' }}" to mean "first apply {{mvar|f}}, then apply {{mvar|g}}" was too confusing and decided to change notations. They write "{{math|''xf'' }}" for "{{math|''f''(''x'')}}" and "{{math|(''xf'' )''g''}}" for "{{math|''g''(''f''(''x''))}}". This can be more natural and seem simpler than writing [[prefix notation|functions on the left]] in some areas – in [[linear algebra]], for instance, when {{mvar|x}} is a [[row vector]] and {{mvar|f}} and {{mvar|g}} denote [[matrix (mathematics)|matrices]] and the composition is by [[matrix multiplication]]. This alternative notation is called [[postfix notation]]. The order is important because matrix multiplication is non-commutative. 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 do {{mvar|f}} then do {{mvar|g}}, in keeping with the order the symbols occur in postfix notation, thus making the notation "{{math|''fg''}}" ambiguous. Computer scientists may write "''f'';''g''" for this, thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in the [[Z notation]] the {{unichar|2A1F|fat [[semicolon]]}} character is used for left [[relation composition]]. 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 {{mvar|g}}, the '''composition operator''' {{math|''C''<sub>''g''</sub>}} is defined as that [[Operator (mathematics)|operator]] which maps functions to functions as
| |
| | |
| ::<math>C_g f = f \circ g.</math>
| |
| | |
| Composition operators are studied in the field of [[operator theory]].
| |
| | |
| == Generalizations ==
| |
| The structures given by composition are axiomatized and generalized in [[category theory]] with the concept of [[morphism]] as the category-theoretical replacement of functions.
| |
| | |
| [[Composition of relations]] is a generalization to [[Relation (mathematics)|relations]], which gives the formula for {{math|''g'' ∘ ''f'' ⊆ ''X'' × ''Z''}} in terms of
| |
| {{math|''f'' ⊆ ''X'' × ''Y''}} and {{math|''g'' ⊆ ''Y'' × ''Z''}} by applying the [[existential quantification]]. Considering functions as its special case (namely [[functional relation]]s), composition of functions satisfies the definition for relations.
| |
| | |
| Composition is possible for multivariate functions. The function resulting when some argument {{math|''x''<sub>i</sub>}} of the function {{math|''f''}} is replaced by the function {{math|''g''}} is called a composition of {{math|''f''}} and {{math|''g''}}, and is denoted {{math|1=''f'' {{!}}<sub>''x''<sub>i</sub> = ''g''</sub>}}
| |
| :<math>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 ''b'', composition degenerates into a (partial) [[Valuation (logic)|valuation]], whose result is also known as '''restriction''' or '''co-factor'''<ref>{{cite journal |author=Bryant, 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 |url=http://www.cs.cmu.edu/~bryant/pubdir/ieeetc86.pdf}}
| |
| </ref>
| |
| | |
| :<math>f|_{x_i = b} = f (x_1, \ldots, x_{i-1}, b, x_{i+1}, \ldots, x_n).</math> | |
| | |
| ==See also==
| |
| * [[Combinatory logic]]
| |
| * [[Function composition (computer science)]]
| |
| * [[Functional decomposition]]
| |
| * [[Flow (mathematics)]]
| |
| * [[Higher-order function]]
| |
| * [[Cobweb plot]] – a graphical technique for functional composition
| |
| * [[Lambda calculus]]
| |
| * [[Functional square root]]
| |
| * [[Fractional calculus]]
| |
| * [[Composition ring]], a formal axiomatization of the composition operation
| |
| | |
| == References ==
| |
| <references />
| |
| | |
| ==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]]
| |