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Inmathematics, thecomposition operator takes twofunctions, and, and returns a new function. Thus, the functiong isapplied after applyingf tox. is pronounced "the composition ofg andf".[1]
Reverse composition applies the operation in the opposite order, applying first and second. Intuitively, reverse composition is a chaining process in which the output of functionf feeds the input of functiong.
The composition of functions is a special case of thecomposition of relations, sometimes also denoted by. As a result, all properties of composition of relations are true of composition of functions,[2] such asassociativity.
The composition of functions is alwaysassociative—a property inherited from thecomposition of relations.[2] That is, iff,g, andh are composable, thenf ∘ (g ∘ h) = (f ∘ g) ∘h.[3] Since the parentheses do not change the result, they are generally omitted.
In a strict sense, the compositiong ∘ f is only meaningful if the codomain off equals the domain ofg; in a wider sense, it is sufficient that the former be an impropersubset of the latter.[nb 1]Moreover, it is often convenient to tacitly restrict the domain off, such thatf produces only values in the domain ofg. For example, the compositiong ∘ f of the functionsf :R →(−∞,+9] defined byf(x) = 9 −x2 andg :[0,+∞) →R defined by can be defined on theinterval[−3,+3].
The functionsg andf are said tocommute with each other ifg ∘ f =f ∘ g. Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example,|x| + 3 = |x + 3| only whenx ≥ 0. The picture shows another example.
The composition ofone-to-one (injective) functions is always one-to-one. Similarly, the composition ofonto (surjective) functions is always onto. It follows that the composition of twobijections is also a bijection. Theinverse function of a composition (assumed invertible) has the property that(f ∘ g)−1 =g−1∘f−1.[4]
Derivatives of compositions involving differentiable functions can be found using thechain rule.Higher derivatives of such functions are given byFaà di Bruno's formula.[3]
Composition of functions is sometimes described as a kind ofmultiplication on a function space, but has very different properties frompointwise multiplication of functions (e.g. composition is notcommutative).[5]
Suppose one has two (or more) functionsf:X →X,g:X →X having the same domain and codomain; these are often calledtransformations. Then one can form chains of transformations composed together, such asf ∘f ∘g ∘f. Such chains have thealgebraic structure of amonoid, called atransformation monoid or (much more seldom) acomposition monoid. In general, transformation monoids can have remarkably complicated structure. One particular notable example is thede Rham curve. The set ofall functionsf:X →X is called thefull transformation semigroup[6] orsymmetric semigroup[7] on X. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.[8])
If the given transformations arebijective (and thus invertible), then the set of all possible combinations of these functions forms atransformation group (also known as apermutation group); and one says that the group isgenerated by these functions.
The set of all bijective functionsf:X →X (calledpermutations) forms a group with respect to function composition. This is thesymmetric group, also sometimes called thecomposition 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).[9]
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 aregular semigroup.[10]
IfY⊆X, then may compose with itself; this is sometimes denoted as. That is:
More generally, for anynatural numbern ≥ 2, thenthfunctionalpower can be defined inductively byf n =f ∘f n−1 =f n−1 ∘f, a notation introduced byHans Heinrich Bürmann[citation needed][11][12] andJohn Frederick William Herschel.[13][11][14][12] Repeated composition of such a function with itself is calledfunction iteration.
Note: Iff takes its values in aring (in particular for real or complex-valuedf ), there is a risk of confusion, asf n could also stand for then-fold product of f, e.g.f 2(x) =f(x) ·f(x).[12] For trigonometric functions, usually the latter is meant, at least for positive exponents.[12] For example, intrigonometry, this superscript notation represents standardexponentiation when used withtrigonometric functions:
sin2(x) = sin(x) · sin(x).
However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g.,tan−1 = arctan ≠ 1/tan.
In some cases, when, for a given functionf, the equationg ∘g =f has a unique solutiong, that function can be defined as thefunctional square root off, then written asg =f 1/2.
More generally, whengn =f has a unique solution for some natural numbern > 0, thenf m/n can be defined asgm.
Under additional restrictions, this idea can be generalized so that theiteration count becomes a continuous parameter; in this case, such a system is called aflow, specified through solutions ofSchröder's equation. Iterated functions and flows occur naturally in the study offractals anddynamical systems.
To avoid ambiguity, some mathematicians[citation needed] choose to use∘ to denote the compositional meaning, writingf∘n(x) for then-th iterate of the functionf(x), as in, for example,f∘3(x) meaningf(f(f(x))). For the same purpose,f[n](x) was used byBenjamin Peirce[15][12] whereasAlfred Pringsheim andJules Molk suggestednf(x) instead.[16][12][nb 2]
Many mathematicians, particularly ingroup theory, omit the composition symbol, writinggf forg ∘f.[17]
During the mid-20th century, some mathematicians adoptedpostfix notation, writingxf forf(x) and(xf)g forg(f(x)).[18] This can be more natural thanprefix notation in many cases, such as inlinear algebra whenx is arow vector andf andg denotematrices and the composition is bymatrix 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 "fg", meaning first applyf and then applyg, in keeping with the order the symbols occur in postfix notation, thus making the notation "fg" ambiguous. Computer scientists may write "f ;g" for this,[19] thereby disambiguating the order of composition. To distinguish the left composition operator from a text semicolon, in theZ notation the ⨾ character is used for leftrelation composition.[20] Since all functions arebinary relations, it is correct to use the [fat] semicolon for function composition as well (see the article oncomposition of relations for further details on this notation).
Given a function g, thecomposition operatorCg is defined as thatoperator which maps functions to functions asComposition operators are studied in the field ofoperator theory.
Function composition appears in one form or another in numerousprogramming languages.
Partial composition is possible formultivariate functions. The function resulting when some argumentxi of the functionf is replaced by the functiong is called a composition off andg in some computer engineering contexts, and is denotedf |xi =g
Wheng is a simple constantb, composition degenerates into a (partial) valuation, whose result is also known asrestriction orco-factor.[21]
In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition ofprimitive recursive function. Givenf, an-ary function, andnm-ary functionsg1, ...,gn, the composition off withg1, ...,gn, is them-ary function
This is sometimes called thegeneralized composite orsuperposition off withg1, ...,gn.[22] 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 chosenprojection functions. Hereg1, ...,gn 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.[23]
A set of finitaryoperations on some base setX is called aclone if it contains all projections and is closed under generalized composition. A clone generally contains operations of variousarities.[22] The notion of commutation also finds an interesting generalization in the multivariate case; a functionf of arityn is said to commute with a functiong of aritym iff is ahomomorphism preservingg, and vice versa, that is:[22]
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 calledmedial or entropic.[22]
Composition can be generalized to arbitrarybinary relations.IfR ⊆X×Y andS ⊆Y ×Z are two binary relations, then their composition amounts to
.
Considering a function as a special case of a binary relation (namelyfunctional relations), function composition satisfies the definition for relation composition. A small circleR∘S has been used for theinfix notation of composition of relations, as well as functions. When used to represent composition of functions however, the text sequence is reversed to illustrate the different operation sequences accordingly.
The composition is defined in the same way forpartial functions and Cayley's theorem has its analogue called theWagner–Preston theorem.[24]
Thecategory of sets with functions asmorphisms is the prototypicalcategory. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition.[25] The structures given by composition are axiomatized and generalized incategory theory with the concept ofmorphism as the category-theoretical replacement of functions. The reversed order of composition in the formula(f ∘ g)−1 = (g−1 ∘f −1) applies forcomposition of relations usingconverse relations, and thus ingroup theory. These structures formdagger categories.
The standard "foundation" for mathematics starts withsets 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.
The composition symbol∘ is encoded asU+2218 ∘RING OPERATOR (∘, ∘); see theDegree symbol article for similar-appearing Unicode characters. InTeX, it is written\circ.
[…] §473.Iterated logarithms […] We note here the symbolism used byPringsheim andMolk in their jointEncyclopédie article: "2logb a = logb (logb a), …,k+1logb a = logb (klogb a)."[a] […] §533.John Herschel's notation for inverse functions, sin−1 x, tan−1 x, etc., was published by him in thePhilosophical Transactions of London, for the year 1813. He says (p. 10): "This notation cos.−1 e must not be understood to signify 1/cos. e, but what is usually written thus, arc (cos.=e)." He admits that some authors use cos.m A for (cos. A)m, but he justifies his own notation by pointing out that sinced2 x, Δ3 x, Σ2 x meandd x, ΔΔΔ x, ΣΣ x, we ought to write sin.2 x for sin. sin. x, log.3 x for log. log. log. x. Just as we writed−n V=∫n V, we may write similarly sin.−1 x=arc (sin.=x), log.−1 x.=cx. Some years later Herschel explained that in 1813 he usedfn(x),f−n(x), sin.−1 x, etc., "as he then supposed for the first time. The work of a German Analyst,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−1, 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."[b] […] §535.Persistence of rival notations for inverse function.— […] The use of Herschel's notation underwent a slight change inBenjamin Peirce's books, to remove the chief objection to them; Peirce wrote: "cos[−1] x," "log[−1] x."[c] […] §537.Powers of trigonometric functions.—Three principal notations have been used to denote, say, the square of sin x, namely, (sin x)2, sin x2, sin2 x. The prevailing notation at present is sin2 x, though the first is least likely to be misinterpreted. In the case of sin2 x two interpretations suggest themselves; first, sin x ⋅ sin x; second,[d] sin (sin x). As functions of the last type do not ordinarily present themselves, the danger of misinterpretation is very much less than in case of log2 x, where log x ⋅ log x and log (log x) are of frequent occurrence in analysis. […] The notation sinn x for (sin x)n has been widely used and is now the prevailing one. […]
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