In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, function composition is an operation that takes two
functions and , and produces a function such that . In this operation, the function is
applied to the result of applying the function to . That is, the functions and are composed to yield a function that maps in
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
to in
codomain .
Intuitively, if is a function of , and is a function of , then is a function of . The resulting ''composite'' function is denoted , defined by for all in .
The notation is read as " of ", " after ", " circle ", " round ", " about ", " composed with ", " following ", " then ", or " on ", or "the composition of and ". Intuitively, composing functions is a chaining process in which the output of function feeds the input of function .
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,
such as the property of
associativity.
But composition of functions is different from
multiplication of functions (if defined at all), and has some quite different properties; in particular, composition of functions is not
commutative.
Examples
* Composition of functions on a finite set: If , and , then , as shown in the figure.
* Composition of functions on an
infinite set: If (where is the set of all
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s) is given by and is given by , then:
* If an airplane's altitude at time is , and the air pressure at altitude is , then is the pressure around the plane at time .
Properties
The composition of functions is always
associative—a property inherited from the
composition of relations.
That is, if , , and are composable, then .
Since the parentheses do not change the result, they are generally omitted.
In a strict sense, the composition is only meaningful if the codomain of equals the domain of ; in a wider sense, it is sufficient that the former be a
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of the latter.
Moreover, it is often convenient to tacitly restrict the domain of , such that produces only values in the domain of . For example, the composition of the functions defined by and defined by
can be defined on the
interval .
The functions and are said to
commute
Commute, commutation or commutative may refer to:
* Commuting, the process of travelling between a place of residence and a place of work
Mathematics
* Commutative property, a property of a mathematical operation whose result is insensitive to th ...
with each other if . Commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, only when . The picture shows another example.
The composition of
one-to-one (injective) functions is always one-to-one. Similarly, the composition of
onto (surjective) functions is always onto. It follows that the composition of two
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
s is also a bijection. The
inverse function of a composition (assumed invertible) has the property that .
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of compositions involving differentiable functions can be found using the
chain rule.
Higher derivatives of such functions are given by
Faà di Bruno's formula.
Composition monoids
Suppose one has two (or more) functions having the same domain and codomain; these are often called ''
transformations''. Then one can form chains of transformations composed together, such as . Such chains have the
algebraic structure of a
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
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 is called the
full transformation semigroup or ''symmetric semigroup''
on . (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions.
)
If the transformations are
bijective (and thus invertible), then the set of all possible combinations of these functions forms a
transformation group; and one says that the group is
generated by these functions. A fundamental result in group theory,
Cayley's theorem, essentially says that any group is in fact just a subgroup of a permutation group (up to
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
).
The set of all bijective functions (called
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pro ...
s) 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.
Functional powers
If , then may compose with itself; this is sometimes denoted as . That is:
More generally, for any
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called '' cardinal ...
, the th functional
power can be defined inductively by , a notation introduced by
Hans Heinrich Bürmann and
John Frederick William Herschel
Sir John Frederick William Herschel, 1st Baronet (; 7 March 1792 – 11 May 1871) was an English polymath active as a mathematician, astronomer, chemist, inventor, experimental photographer who invented the blueprint and did botanical ...
.
Repeated composition of such a function with itself is called
iterated function.
* By convention, is defined as the identity map on 's domain, .
* If even and admits an
inverse function , negative functional powers are defined for as the
negated power of the inverse function: .
Note: If takes its values in a
ring (in particular for real or complex-valued ), there is a risk of confusion, as could also stand for the -fold product of , e.g. .
For trigonometric functions, usually the latter is meant, at least for positive exponents.
For example, in
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
, this superscript notation represents standard
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to ...
when used with
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
:
.
However, for negative exponents (especially −1), it nevertheless usually refers to the inverse function, e.g., .
In some cases, when, for a given function , the equation has a unique solution , that function can be defined as the
functional square root
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function is a function satisfying ...
of , then written as .
More generally, when has a unique solution for some natural number , then can be defined as .
Under additional restrictions, this idea can be generalized so that the
iteration count becomes a continuous parameter; in this case, such a system is called a
flow
Flow may refer to:
Science and technology
* Fluid flow, the motion of a gas or liquid
* Flow (geomorphology), a type of mass wasting or slope movement in geomorphology
* Flow (mathematics), a group action of the real numbers on a set
* Flow (psyc ...
, 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 choose to use to denote the compositional meaning, writing for the -th iterate of the function , as in, for example, meaning . For the same purpose, was used by
Benjamin Peirce whereas
Alfred Pringsheim
Alfred Pringsheim (2 September 1850 – 25 June 1941) was a German mathematician and patron of the arts. He was born in Ohlau, Prussian Silesia (now Oława, Poland) and died in Zürich, Switzerland.
Family and academic career
Pringsheim came ...
and
Jules Molk suggested instead.
Alternative notations
Many mathematicians, particularly in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
, omit the composition symbol, writing for .
In the mid-20th century, some mathematicians decided that writing "" to mean "first apply , then apply " was too confusing and decided to change notations. They write "" for "" and "" for "".
This can be more natural and seem simpler than writing
functions on the left in some areas – in
linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrice ...
, for instance, when is a
row vector and and denote
matrices and the composition is by
matrix multiplication. This alternative notation is called
postfix notation. The order is important because function composition is not necessarily commutative (e.g. matrix multiplication). Successive transformations applying and composing to the right agrees with the left-to-right reading sequence.
Mathematicians who use postfix notation may write "", meaning first apply and then apply , in keeping with the order the symbols occur in postfix notation, thus making the notation "" ambiguous. Computer scientists may write "" for this,
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.
Since all functions are
binary relations, it is correct to use the
atsemicolon for function composition as well (see the article on
composition of relations for further details on this notation).
Composition operator
Given a function , the composition operator 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 language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...
s.
Multivariate functions
Partial composition is possible for
multivariate functions. The function resulting when some argument of the function is replaced by the function is called a composition of and in some computer engineering contexts, and is denoted
When is a simple constant , composition degenerates into a (partial) valuation, whose result is also known as
restriction or ''co-factor''.
In general, the composition of multivariate functions may involve several other functions as arguments, as in the definition of
primitive recursive function. Given , a -ary function, and -ary functions , the composition of with , is the -ary function
This is sometimes called the generalized composite or superposition of ''f'' with .
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 In set theory, a projection is one of two closely related types of functions or operations, namely:
* A set-theoretic operation typified by the ''j''th projection map, written \mathrm_, that takes an element \vec = (x_1,\ \ldots,\ x_j,\ \ldots,\ x ...
s. Here can be seen as a single vector/
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
-valued function in this generalized scheme, in which case this is precisely the standard definition of function composition.
A set of finitary
operation
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
s on some base set ''X'' is called a
clone if it contains all projections and is closed under generalized composition. Note that a clone generally contains operations of various
arities.
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
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
preserving ''g'', and vice versa i.e.:
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.
Generalizations
Composition can be generalized to arbitrary
binary relations.
If and are two binary relations, then their composition is the relation defined as .
Considering a function as a special case of a binary relation (namely
functional relation
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional ...
s), function composition satisfies the definition for relation composition. A small circle 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
partial functions and Cayley's theorem has its analogue called the
Wagner–Preston theorem.
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.
The structures given by composition are axiomatized and generalized in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
with the concept of
morphism as the category-theoretical replacement of functions. The reversed order of composition in the formula applies for
composition of relations using
converse relations, and thus in
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
. These structures form
dagger categories.
Typography
The composition symbol is encoded as ; 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
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of com ...
*
Composition ring
In mathematics, a composition ring, introduced in , is a commutative ring (''R'', 0, +, −, ·), possibly without an identity 1 (see non-unital ring), together with an operation
: \circ: R \times R \rightarrow R
such that, for any three e ...
, 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
In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function is a function satisfying ...
*
Higher-order function
*
Infinite compositions of analytic functions
*
Iterated function
*
Lambda calculus
Lambda calculus (also written as ''λ''-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation th ...
Notes
References
External links
* {{springer, title=Composite function, id=p/c024260
*
Composition of Functions by Bruce Atwood, the
Wolfram Demonstrations Project, 2007.
Functions and mappings
Basic concepts in set theory
Binary operations