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
* Do ...
to in
codomain
In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either the ...
.
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
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
, 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
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
.
But composition of functions is different from
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
of functions (if defined at all), and has some quite different properties; in particular, composition of functions is not
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
.
Examples
![Example for a composition of two functions](https://upload.wikimedia.org/wikipedia/commons/3/38/Example_for_a_composition_of_two_functions.svg)
* Composition of functions on a finite set: If , and , then , as shown in the figure.
* Composition of functions on an
infinite set
In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable.
Properties
The set of natural numbers (whose existence is postulated by the axiom of infinity) is infinite. It is the only set th ...
: 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 real ...
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
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
—a property inherited from the
composition of relations
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
.
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 (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), 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 ...
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
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of i ...
(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 ...
s is also a bijection. The
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
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. F ...
s of compositions involving differentiable functions can be found using the
chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
.
Higher 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. F ...
s of such functions are given by
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after , although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French ...
.
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
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
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.
Monoids ...
, called a ''
transformation monoid In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations ( functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transf ...
'' 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
In mathematics, a de Rham curve is a certain type of fractal curve named in honor of Georges de Rham.
The Cantor function, Cesàro curve, Minkowski's question mark function, the Lévy C curve, the blancmange curve, and Koch curve are all special ...
. The set of ''all'' functions is called the
full transformation semigroup In algebra, a transformation semigroup (or composition semigroup) is a collection of transformations ( functions from a set to itself) that is closed under function composition. If it includes the identity function, it is a monoid, called a transfo ...
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.
)
![Academ Example of similarity with ratio square root of 2](https://upload.wikimedia.org/wikipedia/commons/3/33/Academ_Example_of_similarity_with_ratio_square_root_of_2.svg)
If the transformations are
bijective
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 ...
(and thus invertible), then the set of all possible combinations of these functions forms a
transformation group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
; and one says that the group is
generated by these functions. A fundamental result in group theory,
Cayley's theorem
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group is isomorphic to a subgroup of a symmetric group.
More specifically, is isomorphic to a subgroup of the symmetric group \operatorname(G) whose eleme ...
, 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 is ...
).
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 proc ...
s) forms a group with respect to function composition. This is the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group \m ...
, 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 In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of 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 n ...
, the th functional
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
can be defined inductively by , a notation introduced by
Hans Heinrich Bürmann
Hans Heinrich Bürmann (died 21 June 1817, in Mannheim) was a German mathematician and teacher. He ran an "academy of commerce" in Mannheim since 1795 where he used to teach mathematics. He also served as a censor in Mannheim. He was nominated He ...
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 wor ...
.
Repeated composition of such a function with itself is called
iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is ...
.
* By convention, is defined as the identity map on 's domain, .
* If even and admits an
inverse function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X\t ...
, negative functional powers are defined for as the
negated
In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
power of the inverse function: .
Note: If takes its values in a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
(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. T ...
, 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 re ...
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 all ...
:
.
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 fo ...
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, specified through solutions of
Schröder's equation
Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function , find the function such that
Schröder's equation is an eigenvalue equation for the composition operator that send ...
. Iterated functions and flows occur naturally in the study of
fractals
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illus ...
and
dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ...
.
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
Benjamin Peirce (; April 4, 1809 – October 6, 1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philoso ...
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
Jules Molk (8 December 1857 in Strasbourg, France – 7 May 1914 in Nancy) was a French mathematician who worked on elliptic functions.
The French Academy of Sciences awarded him the Prix Binoux for 1913.
He was appointed to the chair of applie ...
suggested instead.
Alternative notations
Many mathematicians, particularly in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, 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 matrices.
...
, for instance, when is a
row vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
and and denote
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
and the composition is by
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
. This alternative notation is called
postfix notation
Reverse Polish notation (RPN), also known as reverse Łukasiewicz notation, Polish postfix notation or simply postfix notation, is a mathematical notation in which operators ''follow'' their operands, in contrast to Polish notation (PN), in whi ...
. 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 Z notation is a formal specification language used for describing and modelling computing systems. It is targeted at the clear specification of computer programs and computer-based systems in general.
History
In 1974, Jean-Raymond Abrial ...
the ⨾ character is used for left
relation composition
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplicat ...
.
Since all functions are
binary relations
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over sets and is a new set of ordered pairs consisting of elements in and in ...
, it is correct to use the
atsemicolon for function composition as well (see the article on
composition of relations
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
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 mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operat ...
.
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 function
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function (mathematics), function with more than one argument of a function, argument, with all arguments being real number, r ...
s. 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
Restriction, restrict or restrictor may refer to:
Science and technology
* restrict, a keyword in the C programming language used in pointer declarations
* Restriction enzyme, a type of enzyme that cleaves genetic material
Mathematics and logi ...
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
In computability theory, a primitive recursive function is roughly speaking a function that can be computed by a computer program whose loops are all "for" loops (that is, an upper bound of the number of iterations of every loop can be determined ...
. 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-Ma ...
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 structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
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
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
can be generalized to arbitrary
binary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
s.
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 ...
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 function
In mathematics, a partial function from a set to a set is a function from a subset of (possibly itself) to . The subset , that is, the domain of viewed as a function, is called the domain of definition of . If equals , that is, if is de ...
s and Cayley's theorem has its analogue called the
Wagner–Preston theorem In group theory, an inverse semigroup (occasionally called an inversion semigroup) ''S'' is a semigroup in which every element ''x'' in ''S'' has a unique ''inverse'' ''y'' in ''S'' in the sense that ''x = xyx'' and ''y = yxy'', i.e. a regular semig ...
.
The
category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets ''A'' and ''B'' are the total functions from ''A'' to ''B'', and the composition of m ...
with functions as
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
s is the prototypical
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
. 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, cate ...
with the concept of
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...
as the category-theoretical replacement of functions. The reversed order of composition in the formula applies for
composition of relations
In the mathematics of binary relations, the composition of relations is the forming of a new binary relation from two given binary relations ''R'' and ''S''. In the calculus of relations, the composition of relations is called relative multiplica ...
using
converse relation
In mathematics, the converse relation, or transpose, of a binary relation is the relation that occurs when the order of the elements is switched in the relation. For example, the converse of the relation 'child of' is the relation 'parent& ...
s, and thus in
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. These structures form
dagger categories.
Typography
The composition symbol is encoded as ; see the
Degree symbol
The degree symbol or degree sign, , is a typographical symbol that is used, among other things, to represent degrees of arc (e.g. in geographic coordinate systems), hours (in the medical field), degrees of temperature or alcohol proof. The symbo ...
article for similar-appearing Unicode characters. In
TeX
Tex may refer to:
People and fictional characters
* Tex (nickname), a list of people and fictional characters with the nickname
* Joe Tex (1933–1982), stage name of American soul singer Joseph Arrington Jr.
Entertainment
* ''Tex'', the Italian ...
, it is written
\circ
.
See also
*
Cobweb plot
A cobweb plot, or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions, such as the logistic map. Using a cobweb plot, it is possible t ...
– 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 comput ...
*
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 el ...
, a formal axiomatization of the composition operation
*
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a f ...
*
Function composition (computer science)
In computer science, function composition is an act or mechanism to combine simple functions to build more complicated ones. Like the usual composition of functions in mathematics, the result of each function is passed as the argument of the next ...
*
Function of random variable, distribution of a function of a random variable
*
Functional decomposition
In mathematics, functional decomposition is the process of resolving a functional relationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts by function composition.
...
*
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 fo ...
*
Higher-order function
In mathematics and computer science, a higher-order function (HOF) is a function that does at least one of the following:
* takes one or more functions as arguments (i.e. a procedural parameter, which is a parameter of a procedure that is itself ...
*
Infinite compositions of analytic functions
In mathematics, infinite compositions of analytic functions (ICAF) offer alternative formulations of analytic continued fractions, series, products and other infinite expansions, and the theory evolving from such compositions may shed light on the ...
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Iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is ...
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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 ...
Notes
References
External links
* {{springer, title=Composite function, id=p/c024260
*
Composition of Functions by Bruce Atwood, the
Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
, 2007.
Functions and mappings
Basic concepts in set theory
Binary operations