Function composition
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In mathematics, function composition is an operation ∘ that takes two functions f and g, and produces a function h = g ∘ f such that h(x) = g(f(x)). In this operation, the function g is applied to the result of applying the function f to x. That is, the functions f : X → Y and g : Y → Z are composed to yield a function that maps x in domain X to g(f(x)) in codomain Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. The resulting composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.[nb 1]
The notation g ∘ f is read as "g of f ", "g after f ", "g circle f ", "g round f ", "g about f ", "g composed with f ", "g following f ", "f then g", or "g on f ", or "the composition of g and f ". Intuitively, composing functions 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 composition of relations, sometimes also denoted by . As a result, all properties of composition of relations are true of composition of functions,[1] such as the property of associativity.
Composition of functions is different from
Examples
- Composition of functions on a finite set: If f = {(1, 1), (2, 3), (3, 1), (4, 2)}, and g = {(1, 2), (2, 3), (3, 1), (4, 2)}, then g ∘ f = {(1, 2), (2, 1), (3, 2), (4, 3)}, as shown in the figure.
- Composition of functions on an infinite set: If f: R → R (where R is the set of all real numbers) is given by f(x) = 2x + 4 and g: R → R is given by g(x) = x3, then: (f ∘ g)(x) = f(g(x)) = f(x3) = 2x3 + 4, and(g ∘ f)(x) = g(f(x)) = g(2x + 4) = (2x + 4)3.
- If an airplane's altitude at time t is a(t), and the air pressure at altitude x is p(x), then (p ∘ a)(t) is the pressure around the plane at time t.
Properties
The composition of functions is always
In a strict sense, the composition g ∘ f is only meaningful if the codomain of f equals the domain of g; in a wider sense, it is sufficient that the former be an improper subset of the latter.[nb 2] Moreover, it is often convenient to tacitly restrict the domain of f, such that f produces only values in the domain of g. For example, the composition g ∘ f of the functions f : R → (−∞,+9] defined by f(x) = 9 − x2 and g : [0,+∞) → R defined by can be defined on the interval [−3,+3].
The functions g and f are said to
The composition of
Composition monoids
Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain; these are often called )
If the transformations are
The set of all bijective functions f: X → X (called permutations) forms a group with respect to function composition. This is the symmetric group, also sometimes called the composition group.
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.[9]
Functional powers
If Y ⊆ X, then f: X→Y may compose with itself; this is sometimes denoted as f 2. That is:
More generally, for any
- By convention, f 0 is defined as the identity map on f 's domain, idX.
- If Y = X and f: X → X admits an inverse function f −1 (sometimes called «minus first iteration»[citation needed]), negative functional powers f −n are defined for n > 0 as the negated power of the inverse function: f −n = (f −1)n.[12][10][11]
Note: If f takes its values in a ring (in particular for real or complex-valued f ), there is a risk of confusion, as f n could also stand for the n-fold product of f, e.g. f 2(x) = f(x) · f(x).[11] For trigonometric functions, usually the latter is meant, at least for positive exponents.[11] For example, in trigonometry, this superscript notation represents standard exponentiation when used with trigonometric 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 function f, the equation g ∘ g = f has a unique solution g, that function can be defined as the functional square root of f, then written as g = f 1/2.
More generally, when gn = f has a unique solution for some natural number n > 0, then f m/n can be defined as gm.
Under additional restrictions, this idea can be generalized so that the
To avoid ambiguity, some mathematicians[citation needed] choose to use ∘ to denote the compositional meaning, writing f∘n(x) for the n-th iterate of the function f(x), as in, for example, f∘3(x) meaning f(f(f(x))). For the same purpose, f[n](x) was used by Benjamin Peirce[14][11] whereas Alfred Pringsheim and Jules Molk suggested nf(x) instead.[15][11][nb 3]
Alternative notations
Many mathematicians, particularly in group theory, omit the composition symbol, writing gf for g ∘ f.[16]
In the mid-20th century, some mathematicians decided that writing "g ∘ f " to mean "first apply f, then apply g" was too confusing and decided to change notations. They write "xf " for "f(x)" and "(xf)g" for "g(f(x))".
Mathematicians who use postfix notation may write "fg", meaning first apply f and then apply g, 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,
Composition operator
Given a function g, the composition operator Cg is defined as that operator which maps functions to functions as
Composition operators are studied in the field of operator theory.
In programming languages
Function composition appears in one form or another in numerous programming languages.
Multivariate functions
Partial composition is possible for
When g is a simple constant b, composition degenerates into a (partial) valuation, whose result is also known as restriction or co-factor.[20]
In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of primitive recursive function. Given f, a n-ary function, and n m-ary functions g1, ..., gn, the composition of f with g1, ..., gn, is the m-ary function
This is sometimes called the generalized composite or superposition of f with g1, ..., gn.
A set of finitary operations on some base set X is called a clone if it contains all projections and is closed under generalized composition. A clone generally contains operations of various arities.[21] 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 i.e.:[21]
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 or entropic.[21]
Generalizations
Composition can be generalized to arbitrary binary relations. If R ⊆ X × Y and S ⊆ Y × Z are two binary relations, then their composition R∘S is the relation defined as {(x, z) ∈ X × Z : ∃y ∈ Y. (x, y) ∈ R ∧ (y, z) ∈ S}. Considering a function as a special case of a binary relation (namely functional relations), function composition satisfies the definition for relation composition. A small circle R∘S has been used for the infix 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 for
The category of sets with functions as morphisms is the prototypical category. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition.[24] 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 (f ∘ g)−1 = (g−1 ∘ f −1) applies for composition of relations using converse relations, and thus in group theory. These structures form dagger categories.
The standard "foundation" for mathematics starts with 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.
Typography
The composition symbol ∘ is encoded as U+2218 ∘ RING OPERATOR (∘, ∘); see the Degree symbol article for similar-appearing Unicode characters. In TeX, it is written \circ
.
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)
- Function of random variable, distribution of a function of a random variable
- Functional decomposition
- Functional square root
- Higher-order function
- Infinite compositions of analytic functions
- Iterated function
- Lambda calculus
Notes
- ISBN 0-387-94599-7.
- inclusion function.
- David Patterson Ellerman's (1995) nx pre-superscript notation for roots.
References
- ^ ISBN 978-1-139-45097-3.
- ^ "3.4: Composition of Functions". Mathematics LibreTexts. 2020-01-16. Retrieved 2020-08-28.
- ^ a b Weisstein, Eric W. "Composition". mathworld.wolfram.com. Retrieved 2020-08-28.
- ISBN 978-0-471-37122-9.
- ISBN 978-1-4704-1493-1.
- ISBN 978-0-8247-9662-4.
- ISBN 978-0-89871-569-9.
- ISBN 978-0-88385-757-1.
- ISBN 978-1-84800-281-4.
- ^ Herschel, John Frederick William (1820). "Part III. Section I. Examples of the Direct Method of Differences". A Collection of Examples of the Applications of the Calculus of Finite Differences. Cambridge, UK: Printed by J. Smith, sold by J. Deighton & sons. pp. 1–13 [5–6]. Archived from the original on 2020-08-04. Retrieved 2020-08-04. [1] (NB. Inhere, Herschel refers to his 1813 work and mentions Hans Heinrich Bürmann's older work.)
- ^ Philosophical 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 since d2 x, Δ3 x, Σ2 x mean dd 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 write d−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 used fn(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 in Benjamin 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. […] (xviii+367+1 pages including 1 addenda page) (NB. ISBN and link for reprint of 2nd edition by Cosimo, Inc., New York, USA, 2013.)
- ^ S2CID 118124706.
- ^ Peano, Giuseppe (1903). Formulaire mathématique (in French). Vol. IV. p. 229.
- ^ Peirce, Benjamin (1852). Curves, Functions and Forces. Vol. I (new ed.). Boston, USA. p. 203.
{{cite book}}
: CS1 maint: location missing publisher (link) - ^ Pringsheim, Alfred; Molk, Jules (1907). Encyclopédie des sciences mathématiques pures et appliquées (in French). Vol. I. p. 195. Part I.
- ISBN 978-0-8218-4808-1.
- ISBN 978-1-4419-8047-2.
- ISBN 978-0-13-120486-7.)
- ^ ISO/IEC 13568:2002(E), p. 23
- S2CID 10385726.
- ^ ISBN 978-1-4398-5129-6.
- ISBN 978-1-118-31533-0.
- ISBN 0-8218-0627-0.
- ISBN 978-0-471-50405-4.
- ^ "Saunders Mac Lane - Quotations". Maths History. Retrieved 2024-02-13.
External links
- "Composite function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- "Composition of Functions" by Bruce Atwood, the Wolfram Demonstrations Project, 2007.