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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 ...
, a function from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the
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 ...
of the function and the set is called the
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 ...
of the function.Codomain ''Encyclopedia of Mathematics'
Codomain. ''Encyclopedia of Mathematics''
/ref> The earliest known approach to the notion of function can be traced back to works of Persian mathematicians
Al-Biruni Abu Rayhan Muhammad ibn Ahmad al-Biruni (973 – after 1050) commonly known as al-Biruni, was a Khwarazmian Iranian in scholar and polymath during the Islamic Golden Age. He has been called variously the "founder of Indology", "Father of Co ...
and
Sharaf al-Din al-Tusi Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī ( fa, شرف‌الدین مظفر بن محمد بن مظفر توسی; 1135 – 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the ...
. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a
planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...
is a ''function'' of time.
Historically History (derived ) is the systematic study and the documentation of the human activity. The time period of event before the invention of writing systems is considered prehistory. "History" is an umbrella term comprising past events as well ...
, the concept was elaborated with the
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
at the end of the 17th century, and, until the 19th century, the functions that were considered were
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
(that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, and this greatly enlarged the domains of application of the concept. A function is most often denoted by letters such as , and , and the value of a function at an element of its domain is denoted by ; the numerical value resulting from the ''function evaluation'' at a particular input value is denoted by replacing with this value; for example, the value of at is denoted by . When the function is not named and is represented by an
expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, o ...
, the value of the function at, say, may be denoted by . For example, the value at of the function that maps to (x+1)^2 may be denoted by \left.(x+1)^2\right\vert_ (which results in A function is uniquely represented by the set of all
pairs Concentration, also known as Memory, Shinkei-suijaku (Japanese meaning "nervous breakdown"), Matching Pairs, Match Match, Match Up, Pelmanism, Pexeso or simply Pairs, is a card game in which all of the cards are laid face down on a surface and tw ...
, called the '' graph of the function'', a popular means of illustrating the function.This definition of "graph" refers to a ''set'' of pairs of objects. Graphs, in the sense of ''diagrams'', are most applicable to functions from the real numbers to themselves. All functions can be described by sets of pairs but it may not be practical to construct a diagram for functions between other sets (such as sets of matrices). When the domain and the codomain are sets of real numbers, each such pair may be thought of as the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
of a point in the plane. Functions are widely used in
science Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for ...
,
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.


Definition

A function from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
to a set is an assignment of an element of to each element of . The set is called the
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 ...
of the function and the set is called the
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 ...
of the function. A function, its domain, and its codomain, are declared by the notation , and the value of a function at an element of , denoted by , is called the ''image'' of under , or the ''value'' of applied to the ''argument'' . Functions are also called ''
maps A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
'' or ''mappings'', though some authors make some distinction between "maps" and "functions" (see ). Two functions and are equal if their domain and codomain sets are the same and their output values agree on the whole domain. More formally, given and , we have if and only if for all .This follows from the
axiom of extensionality In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements ...
, which says two sets are the same if and only if they have the same members. Some authors drop codomain from a definition of a function, and in that definition, the notion of equality has to be handled with care; see, for example,
The domain and codomain are not always explicitly given when a function is defined, and, without some (possibly difficult) computation, one might only know that the domain is contained in a larger set. Typically, this occurs in
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, where "a function often refers to a function that may have a proper subsetcalled the ''domain of definition'' by some authors, notably computer science of as domain. For example, a "function from the reals to the reals" may refer to a
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
function of a real variable. However, a "function from the reals to the reals" does not mean that the domain of the function is the whole set of the
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, but only that the domain is a set of real numbers that contains a non-empty
open interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
. Such a function is then called a
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 ...
. For example, if is a function that has the real numbers as domain and codomain, then a function mapping the value to the value is a function from the reals to the reals, whose domain is the set of the reals , such that . The
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
or
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of a function is the set of the
images An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of all elements in the domain.


Total, univalent relation

Any subset of the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of two sets and defines a
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 ...
between these two sets. It is immediate that an arbitrary relation may contain pairs that violate the necessary conditions for a function given above. A binary relation is univalent (also called right-unique) if :\forall x\in X, \forall y\in Y, \forall z\in Y, \quad ((x,y)\in R \land (x,z)\in R)\implies y=z. A binary relation is total if :\forall x\in X, \exists y\in Y, \quad(x,y)\in R. A
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 ...
is a binary relation that is univalent, and a function is a binary relation that is univalent and total. Various properties of functions and function composition may be reformulated in the language of relations.
Gunther Schmidt Gunther Schmidt (born 1939, Rüdersdorf) is a German mathematician who works also in informatics. Life Schmidt began studying Mathematics in 1957 at Göttingen University. His academic teachers were in particular Kurt Reidemeister, Wilhelm K ...
( 2011) ''Relational Mathematics'', Encyclopedia of Mathematics and its Applications, vol. 132, sect 5.1 Functions, pp. 49–60,
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...
For example, a function is
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
if the
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& ...
is univalent, where the converse relation is defined as


Set exponentiation

The set of all functions from a set X to a set Y is commonly denoted as :Y^X, which is read as Y ''to the power'' X. This notation is the same as the notation for the
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
of a
family Family (from la, familia) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). The purpose of the family is to maintain the well-being of its ...
of copies of Y indexed by X: :Y^X=\prod_Y. The identity of these two notations is motivated by the fact that a function f can be identified with the element of the Cartesian product such that the component of index x is f(x). When Y has two elements, Y^X is commonly denoted 2^X and called the
powerset In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postu ...
of . It can be identified with the set of all
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 ...
s of X, through the one-to-one correspondence that associates to each subset S\subseteq X the function f such that f(x)=1 if x\in S and f(x)=0 otherwise.


Notation

There are various standard ways for denoting functions. The most commonly used notation is functional notation, which is the first notation described below.


Functional notation

In functional notation, the function is immediately given a name, such as , and its definition is given by what does to the explicit argument , using a formula in terms of . For example, the function which takes a real number as input and outputs that number plus 1 is denoted by :f(x)=x+1. If a function is defined in this notation, its domain and codomain are implicitly taken to both be \R, the set of real numbers. If the formula cannot be evaluated at all real numbers, then the domain is implicitly taken to be the maximal subset of \R on which the formula can be evaluated; see
Domain of a function In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by \operatorname(f) or \operatornamef, where is the function. More precisely, given a function f\colon X\to Y, the domain of is . ...
. A more complicated example is the function :f(x)=\sin(x^2+1). In this example, the function takes a real number as input, squares it, then adds 1 to the result, then takes the sine of the result, and returns the final result as the output. When the symbol denoting the function consists of several characters and no ambiguity may arise, the parentheses of functional notation might be omitted. For example, it is common to write instead of . Functional notation was first used by
Leonhard Euler Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
in 1734. Some widely used functions are represented by a symbol consisting of several letters (usually two or three, generally an abbreviation of their name). In this case, a
roman type In Latin script typography, roman is one of the three main kinds of historical type, alongside blackletter and italic. Roman type was modelled from a European scribal manuscript style of the 15th century, based on the pairing of inscriptional ...
is customarily used instead, such as "" for the
sine function In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
, in contrast to italic font for single-letter symbols. When using this notation, one often encounters the
abuse of notation In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors an ...
whereby the notation can refer to the value of at , or to the function itself. If the variable was previously declared, then the notation unambiguously means the value of at . Otherwise, it is useful to understand the notation as being both simultaneously; this allows one to denote composition of two functions and in a succinct manner by the notation . However, distinguishing and can become important in cases where functions themselves serve as inputs for other functions. (A function taking another function as an input is termed a ''
functional 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 sy ...
''.) Other approaches of notating functions, detailed below, avoid this problem but are less commonly used.


Arrow notation

Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. For example, x\mapsto x+1 is the function which takes a real number as input and outputs that number plus 1. Again a domain and codomain of \R is implied. The domain and codomain can also be explicitly stated, for example: :\begin \operatorname\colon \Z &\to \Z\\ x &\mapsto x^2.\end This defines a function from the integers to the integers that returns the square of its input. As a common application of the arrow notation, suppose f\colon X\times X\to Y;\;(x,t) \mapsto f(x,t) is a function in two variables, and we want to refer to a partially applied function X\to Y produced by fixing the second argument to the value without introducing a new function name. The map in question could be denoted x\mapsto f(x,t_0) using the arrow notation. The expression x\mapsto f(x,t_0) (read: "the map taking to ") represents this new function with just one argument, whereas the expression refers to the value of the function at the


Index notation

Index notation is often used instead of functional notation. That is, instead of writing , one writes f_x. This is typically the case for functions whose domain is the set of the
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 ...
s. Such a function is called a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
, and, in this case the element f_n is called the th element of the sequence. The index notation is also often used for distinguishing some variables called
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s from the "true variables". In fact, parameters are specific variables that are considered as being fixed during the study of a problem. For example, the map x\mapsto f(x,t) (see above) would be denoted f_t using index notation, if we define the collection of maps f_t by the formula f_t(x)=f(x,t) for all x,t\in X.


Dot notation

In the notation x\mapsto f(x), the symbol does not represent any value, it is simply a
placeholder Placeholder may refer to: Language * Placeholder name, a term or terms referring to something or somebody whose name is not known or, in that particular context, is not significant or relevant. * Filler text, text generated to fill space or provi ...
meaning that, if is replaced by any value on the left of the arrow, it should be replaced by the same value on the right of the arrow. Therefore, may be replaced by any symbol, often an
interpunct An interpunct , also known as an interpoint, middle dot, middot and centered dot or centred dot, is a punctuation mark consisting of a vertically centered dot used for interword separation in ancient Latin script. (Word-separating spaces did no ...
"". This may be useful for distinguishing the function from its value at . For example, a(\cdot)^2 may stand for the function x\mapsto ax^2, and \int_a^ f(u)\,du may stand for a function defined by an integral with variable upper bound: x\mapsto \int_a^x f(u)\,du.


Specialized notations

There are other, specialized notations for functions in sub-disciplines of mathematics. For example, 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. ...
and
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
,
linear form In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the s ...
s and the vectors they act upon are denoted using a
dual pair In mathematics, a dual system, dual pair, or duality over a field \mathbb is a triple (X, Y, b) consisting of two vector spaces X and Y over \mathbb and a non-degenerate bilinear map b : X \times Y \to \mathbb. Duality theory, the study of dual ...
to show the underlying duality. This is similar to the use of
bra–ket notation In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets". A ket is of the form , v \rangle. Mathema ...
in quantum mechanics. In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
and the
theory of computation In theoretical computer science and mathematics, the theory of computation is the branch that deals with what problems can be solved on a model of computation, using an algorithm, how efficiently they can be solved or to what degree (e.g., a ...
, the function notation of
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 ...
is used to explicitly express the basic notions of function
abstraction Abstraction in its main sense is a conceptual process wherein general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods. "An abstr ...
and application. 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 ...
and
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, networks of functions are described in terms of how they and their compositions
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 using
commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...
s that extend and generalize the arrow notation for functions described above.


Other terms

A function is often also called a map or a mapping, but some authors make a distinction between the term "map" and "function". For example, the term "map" is often reserved for a "function" with some sort of special structure (e.g.
maps of manifolds In mathematics, more specifically in differential geometry and topology, various types of functions between manifolds are studied, both as objects in their own right and for the light they shed Types of maps Just as there are various types of m ...
). In particular ''map'' is often used in place of ''homomorphism'' for the sake of succinctness (e.g.,
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
or ''map from to '' instead of ''
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) wh ...
from to ''). Some authors reserve the word ''mapping'' for the case where the structure of the codomain belongs explicitly to the definition of the function. Some authors, such as
Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...
, use "function" only to refer to maps for which the
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 ...
is a subset of the
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
numbers, and use the term ''mapping'' for more general functions. In the theory of
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s, a map denotes an
evolution function 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 ...
used to create discrete dynamical systems. See also
Poincaré map In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is the intersection of a periodic orbit in the state space of a continuous dynamical system with a certain lower-dimensiona ...
. Whichever definition of ''map'' is used, related terms like ''
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 ...
'', ''
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 ...
'', ''
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
'', ''
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
'' have the same meaning as for a function.


Specifying a function

Given a function f, by definition, to each element x of the domain of the function f, there is a unique element associated to it, the value f(x) of f at x. There are several ways to specify or describe how x is related to f(x), both explicitly and implicitly. Sometimes, a theorem or an
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
asserts the existence of a function having some properties, without describing it more precisely. Often, the specification or description is referred to as the definition of the function f.


By listing function values

On a finite set, a function may be defined by listing the elements of the codomain that are associated to the elements of the domain. For example, if A = \, then one can define a function f\colon A \to \mathbb by f(1) = 2, f(2) = 3, f(3) = 4.


By a formula

Functions are often defined by a
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
that describes a combination of
arithmetic operations Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th cen ...
and previously defined functions; such a formula allows computing the value of the function from the value of any element of the domain. For example, in the above example, f can be defined by the formula f(n) = n+1, for n\in\. When a function is defined this way, the determination of its domain is sometimes difficult. If the formula that defines the function contains divisions, the values of the variable for which a denominator is zero must be excluded from the domain; thus, for a complicated function, the determination of the domain passes through the computation of the zeros of auxiliary functions. Similarly, if
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
s occur in the definition of a function from \mathbb to \mathbb, the domain is included in the set of the values of the variable for which the arguments of the square roots are nonnegative. For example, f(x)=\sqrt defines a function f\colon \mathbb \to \mathbb whose domain is \mathbb, because 1+x^2 is always positive if is a real number. On the other hand, f(x)=\sqrt defines a function from the reals to the reals whose domain is reduced to the interval . (In old texts, such a domain was called the ''domain of definition'' of the function.) Functions are often classified by the nature of formulas that define them: *A
quadratic function In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
is a function that may be written f(x) = ax^2+bx+c, where are constants. *More generally, a
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
is a function that can be defined by a formula involving only additions, subtractions, multiplications, and
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 ...
to nonnegative integers. For example, f(x) = x^3-3x-1, and f(x) = (x-1)(x^3+1) +2x^2 -1. *A
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
is the same, with divisions also allowed, such as f(x) = \frac, and f(x) = \frac 1+\frac 3x-\frac 2. *An
algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additio ...
is the same, with th roots and roots of polynomials also allowed. *An
elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
Here "elementary" has not exactly its common sense: although most functions that are encountered in elementary courses of mathematics are elementary in this sense, some elementary functions are not elementary for the common sense, for example, those that involve roots of polynomials of high degree. is the same, with
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
s and
exponential functions Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above *Exponential decay, decrease at a rate proportional to value *Expo ...
allowed.


Inverse and implicit functions

A function f\colon X\to Y, with domain and codomain , is
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 ...
, if for every in , there is one and only one element in such that . In this case, 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 is the function f^\colon Y \to X that maps y\in Y to the element x\in X such that . For example, the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
, that maps the real numbers onto the positive numbers. If a function f\colon X\to Y is not bijective, it may occur that one can select subsets E\subseteq X and F\subseteq Y such that the
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 ...
of to is a bijection from to , and has thus an inverse. The
inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
are defined this way. For example, the
cosine function In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...
induces, by restriction, a bijection from the interval onto the interval , and its inverse function, called
arccosine In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
, maps onto . The other inverse trigonometric functions are defined similarly. More generally, given a
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 ...
between two sets and , let be a subset of such that, for every x\in E, there is some y\in Y such that . If one has a criterion allowing selecting such an for every x\in E, this defines a function f\colon E\to Y, called an
implicit function In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit functi ...
, because it is implicitly defined by the relation . For example, the equation of the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucl ...
x^2+y^2=1 defines a relation on real numbers. If there are two possible values of , one positive and one negative. For , these two values become both equal to 0. Otherwise, there is no possible value of . This means that the equation defines two implicit functions with domain and respective codomains and . In this example, the equation can be solved in , giving y=\pm \sqrt, but, in more complicated examples, this is impossible. For example, the relation y^5+y+x=0 defines as an implicit function of , called the
Bring radical In algebra, the Bring radical or ultraradical of a real number ''a'' is the unique real root of the polynomial : x^5 + x + a. The Bring radical of a complex number ''a'' is either any of the five roots of the above polynomial (it is thus m ...
, which has \mathbb R as domain and range. The Bring radical cannot be expressed in terms of the four arithmetic operations and th roots. The implicit function theorem provides mild
differentiability In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
conditions for existence and uniqueness of an implicit function in the neighborhood of a point.


Using differential calculus

Many functions can be defined as the
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
of another function. This is the case of the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
, which is the antiderivative of that is 0 for . Another common example is the
error function In mathematics, the error function (also called the Gauss error function), often denoted by , is a complex function of a complex variable defined as: :\operatorname z = \frac\int_0^z e^\,\mathrm dt. This integral is a special (non-elementary ...
. More generally, many functions, including most
special function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by ...
s, can be defined as solutions of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. The simplest example is probably the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
, which can be defined as the unique function that is equal to its derivative and takes the value 1 for .
Power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
can be used to define functions on the domain in which they converge. For example, the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
is given by e^x = \sum_^ . However, as the coefficients of a series are quite arbitrary, a function that is the sum of a convergent series is generally defined otherwise, and the sequence of the coefficients is the result of some computation based on another definition. Then, the power series can be used to enlarge the domain of the function. Typically, if a function for a real variable is the sum of its
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
in some interval, this power series allows immediately enlarging the domain to a subset of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, the
disc of convergence In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or \infty. When it is positive, the power series co ...
of the series. Then
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
allows enlarging further the domain for including almost the whole
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. This process is the method that is generally used for defining the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...
, the
exponential Exponential may refer to any of several mathematical topics related to exponentiation, including: *Exponential function, also: **Matrix exponential, the matrix analogue to the above * Exponential decay, decrease at a rate proportional to value *Exp ...
and the
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 ...
of a complex number.


By recurrence

Functions whose domain are the nonnegative integers, known as
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s, are often defined by
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s. The
factorial In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \t ...
function on the nonnegative integers (n\mapsto n!) is a basic example, as it can be defined by the recurrence relation :n!=n(n-1)!\quad\text\quad n>0, and the initial condition :0!=1.


Representing a function

A
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
is commonly used to give an intuitive picture of a function. As an example of how a graph helps to understand a function, it is easy to see from its graph whether a function is increasing or decreasing. Some functions may also be represented by
bar chart A bar chart or bar graph is a chart or graph that presents categorical data with rectangular bars with heights or lengths proportional to the values that they represent. The bars can be plotted vertically or horizontally. A vertical bar chart is ...
s.


Graphs and plots

Given a function f\colon X\to Y, its ''graph'' is, formally, the set :G=\. In the frequent case where and are subsets of the
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 (or may be identified with such subsets, e.g.
intervals Interval may refer to: Mathematics and physics * Interval (mathematics), a range of numbers ** Partially ordered set#Intervals, its generalization from numbers to arbitrary partially ordered sets * A statistical level of measurement * Interval e ...
), an element (x,y)\in G may be identified with a point having coordinates in a 2-dimensional coordinate system, e.g. the
Cartesian plane A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
. Parts of this may create a
plot Plot or Plotting may refer to: Art, media and entertainment * Plot (narrative), the story of a piece of fiction Music * ''The Plot'' (album), a 1976 album by jazz trumpeter Enrico Rava * The Plot (band), a band formed in 2003 Other * ''Plot' ...
that represents (parts of) the function. The use of plots is so ubiquitous that they too are called the ''graph of the function''. Graphic representations of functions are also possible in other coordinate systems. For example, the graph of the
square function In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power  2, and is denoted by a superscript 2; for instance, the square ...
:x\mapsto x^2, consisting of all points with coordinates (x, x^2) for x\in \R, yields, when depicted in Cartesian coordinates, the well known
parabola In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...
. If the same quadratic function x\mapsto x^2, with the same formal graph, consisting of pairs of numbers, is plotted instead in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
(r,\theta) =(x,x^2), the plot obtained is
Fermat's spiral A Fermat's spiral or parabolic spiral is a plane curve with the property that the area between any two consecutive full turns around the spiral is invariant. As a result, the distance between turns grows in inverse proportion to their distance f ...
.


Tables

A function can be represented as a table of values. If the domain of a function is finite, then the function can be completely specified in this way. For example, the multiplication function f\colon\^2 \to \mathbb defined as f(x,y)=xy can be represented by the familiar
multiplication table In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essenti ...
On the other hand, if a function's domain is continuous, a table can give the values of the function at specific values of the domain. If an intermediate value is needed,
interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a n ...
can be used to estimate the value of the function. For example, a portion of a table for the sine function might be given as follows, with values rounded to 6 decimal places: Before the advent of handheld calculators and personal computers, such tables were often compiled and published for functions such as logarithms and trigonometric functions.


Bar chart

Bar charts are often used for representing functions whose domain is a finite set, the
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 ...
s, or the
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. In this case, an element of the domain is represented by an interval of the -axis, and the corresponding value of the function, , is represented by a
rectangle In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
whose base is the interval corresponding to and whose height is (possibly negative, in which case the bar extends below the -axis).


General properties

This section describes general properties of functions, that are independent of specific properties of the domain and the codomain.


Standard functions

There are a number of standard functions that occur frequently: * For every set , there is a unique function, called the , or empty map, from the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
to . The graph of an empty function is the empty set.By definition, the graph of the empty function to is a subset of the Cartesian product , and this product is empty. The existence of empty functions is needed both for the coherency of the theory and for avoiding exceptions concerning the empty set in many statements. Under the usual set-theoretic definition of a function as an ordered triplet (or equivalent ones), there is exactly one empty function for each set, thus the empty function \varnothing \mapsto X is not equal to \varnothing \mapsto Y if and only if X\ne Y, although their graph are both the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
. * For every set and every
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
, there is a unique function from to , which maps every element of to . This is a surjection (see below) unless is the empty set. * Given a function f\colon X\to Y, the ''canonical surjection'' of onto its image f(X)=\ is the function from to that maps to . * For every
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 a set , the
inclusion map In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element x of A to x, treated as an element of B: \iota : A\rightarrow B, \qquad \iot ...
of into is the injective (see below) function that maps every element of to itself. * The
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on a set , often denoted by , is the inclusion of into itself.


Function composition

Given two functions f\colon X\to Y and g\colon Y\to Z such that the domain of is the codomain of , their ''composition'' is the function g \circ f\colon X \rightarrow Z defined by :(g \circ f)(x) = g(f(x)). That is, the value of g \circ f is obtained by first applying to to obtain and then applying to the result to obtain . In the notation the function that is applied first is always written on the right. The composition g\circ f is an
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 ...
on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both g \circ f and f \circ g satisfy these conditions, the composition is not necessarily
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 ...
, that is, the functions g \circ f and f \circ g need not be equal, but may deliver different values for the same argument. For example, let and , then g(f(x))=x^2+1 and f(g(x)) = (x+1)^2 agree just for x=0. The function composition is
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 ...
in the sense that, if one of (h\circ g)\circ f and h\circ (g\circ f) is defined, then the other is also defined, and they are equal. Thus, one writes :h\circ g\circ f = (h\circ g)\circ f = h\circ (g\circ f). The
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
s \operatorname_X and \operatorname_Y are respectively a
right identity In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
and a left identity for functions from to . That is, if is a function with domain , and codomain , one has f\circ \operatorname_X = \operatorname_Y \circ f = f. File:Function machine5.svg, A composite function ''g''(''f''(''x'')) can be visualized as the combination of two "machines". File:Example for a composition of two functions.svg, A simple example of a function composition File:Compfun.svg, Another composition. In this example, .


Image and preimage

Let f\colon X\to Y. The ''image'' under of an element of the domain is . If is any subset of , then the ''image'' of under , denoted , is the subset of the codomain consisting of all images of elements of , that is, :f(A)=\. The ''image'' of is the image of the whole domain, that is, . It is also called the
range Range may refer to: Geography * Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra) ** Mountain range, a group of mountains bordered by lowlands * Range, a term used to i ...
of , although the term ''range'' may also refer to the codomain.''Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology'', p. 15. ISO 80000-2 (ISO/IEC 2009-12-01) On the other hand, the ''
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
'' or ''
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
'' under of an element of the codomain is the set of all elements of the domain whose images under equal . In symbols, the preimage of is denoted by f^(y) and is given by the equation :f^(y) = \. Likewise, the preimage of a subset of the codomain is the set of the preimages of the elements of , that is, it is the subset of the domain consisting of all elements of whose images belong to . It is denoted by f^(B) and is given by the equation :f^(B) = \. For example, the preimage of \ under the
square function In mathematics, a square is the result of multiplying a number by itself. The verb "to square" is used to denote this operation. Squaring is the same as raising to the power  2, and is denoted by a superscript 2; for instance, the square ...
is the set \. By definition of a function, the image of an element of the domain is always a single element of the codomain. However, the preimage f^(y) of an element of the codomain may be
empty Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...
or contain any number of elements. For example, if is the function from the integers to themselves that maps every integer to 0, then f^(0) = \mathbb. If f\colon X\to Y is a function, and are subsets of , and and are subsets of , then one has the following properties: * A\subseteq B \Longrightarrow f(A)\subseteq f(B) * C\subseteq D \Longrightarrow f^(C)\subseteq f^(D) * A \subseteq f^(f(A)) * C \supseteq f(f^(C)) * f(f^(f(A)))=f(A) * f^(f(f^(C)))=f^(C) The preimage by of an element of the codomain is sometimes called, in some contexts, the
fiber Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
of under . If a function has an inverse (see below), this inverse is denoted f^. In this case f^(C) may denote either the image by f^ or the preimage by of . This is not a problem, as these sets are equal. The notation f(A) and f^(C) may be ambiguous in the case of sets that contain some subsets as elements, such as \. In this case, some care may be needed, for example, by using square brackets f f^ /math> for images and preimages of subsets and ordinary parentheses for images and preimages of elements.


Injective, surjective and bijective functions

Let f\colon X\to Y be a function. The function is ''
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
'' (or ''one-to-one'', or is an ''injection'') if for any two different elements and of . Equivalently, is injective if and only if, for any y\in Y, the preimage f^(y) contains at most one element. An empty function is always injective. If is not the empty set, then is injective if and only if there exists a function g\colon Y\to X such that g\circ f=\operatorname_X, that is, if has a left inverse. ''Proof'': If is injective, for defining , one chooses an element x_0 in (which exists as is supposed to be nonempty),The
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
is not needed here, as the choice is done in a single set.
and one defines by g(y)=x if y=f(x) and g(y)=x_0 if y\not\in f(X). Conversely, if g\circ f=\operatorname_X, and y=f(x), then x=g(y), and thus f^(y)=\. The function is ''
surjective 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 ...
'' (or ''onto'', or is a ''surjection'') if its range f(X) equals its codomain Y, that is, if, for each element y of the codomain, there exists some element x of the domain such that f(x) = y (in other words, the preimage f^(y) of every y\in Y is nonempty). If, as usual in modern mathematics, the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
is assumed, then is surjective if and only if there exists a function g\colon Y\to X such that f\circ g=\operatorname_Y, that is, if has a right inverse. The axiom of choice is needed, because, if is surjective, one defines by g(y)=x, where x is an ''arbitrarily chosen'' element of f^(y). The function is ''
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 ...
'' (or is a ''bijection'' or a ''one-to-one correspondence'') if it is both injective and surjective. That is, is bijective if, for any y\in Y, the preimage f^(y) contains exactly one element. The function is bijective if and only if it 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 ...
, that is, a function g\colon Y\to X such that g\circ f=\operatorname_X and f\circ g=\operatorname_Y. (Contrarily to the case of surjections, this does not require the axiom of choice; the proof is straightforward). Every function f\colon X\to Y may be factorized as the composition i\circ s of a surjection followed by an injection, where is the canonical surjection of onto and is the canonical injection of into . This is the ''canonical factorization'' of . "One-to-one" and "onto" are terms that were more common in the older English language literature; "injective", "surjective", and "bijective" were originally coined as French words in the second quarter of the 20th century by the
Bourbaki group Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure - PSL (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in ...
and imported into English. As a word of caution, "a one-to-one function" is one that is injective, while a "one-to-one correspondence" refers to a bijective function. Also, the statement " maps ''onto'' " differs from " maps ''into'' ", in that the former implies that is surjective, while the latter makes no assertion about the nature of . In a complicated reasoning, the one letter difference can easily be missed. Due to the confusing nature of this older terminology, these terms have declined in popularity relative to the Bourbakian terms, which have also the advantage of being more symmetrical.


Restriction and extension

If f\colon X \to Y is a function and ''S'' is a subset of ''X'', then the ''restriction'' of f to ''S'', denoted f, _S, is the function from ''S'' to ''Y'' defined by :f, _S(x) = f(x) for all ''x'' in ''S''. Restrictions can be used to define partial
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 ...
s: if there is 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 ...
''S'' of the domain of a function f such that f, _S is injective, then the canonical surjection of f, _S onto its image f, _S(S) = f(S) is a bijection, and thus has an inverse function from f(S) to ''S''. One application is the definition of
inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
. For example, the cosine function is injective when restricted to the interval . The image of this restriction is the interval , and thus the restriction has an inverse function from to , which is called
arccosine In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
and is denoted . Function restriction may also be used for "gluing" functions together. Let X=\bigcup_U_i be the decomposition of as a
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
of subsets, and suppose that a function f_i\colon U_i \to Y is defined on each U_i such that for each pair i, j of indices, the restrictions of f_i and f_j to U_i \cap U_j are equal. Then this defines a unique function f\colon X \to Y such that f, _ = f_i for all . This is the way that functions on
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s are defined. An ''extension'' of a function is a function such that is a restriction of . A typical use of this concept is the process of
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
, that allows extending functions whose domain is a small part of the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
to functions whose domain is almost the whole complex plane. Here is another classical example of a function extension that is encountered when studying homographies of the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
. A ''homography'' is a function h(x)=\frac such that . Its domain 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 different from -d/c, and its image is the set of all real numbers different from a/c. If one extends the real line to the
projectively extended real line In real analysis, the projectively extended real line (also called the one-point compactification of the real line), is the extension of the set of the real numbers, \mathbb, by a point denoted . It is thus the set \mathbb\cup\ with the standar ...
by including , one may extend to a bijection from the extended real line to itself by setting h(\infty)=a/c and h(-d/c)=\infty.


Multivariate function

A multivariate function, or function of several variables is a function that depends on several arguments. Such functions are commonly encountered. For example, the position of a car on a road is a function of the time travelled and its average speed. More formally, a function of variables is a function whose domain is a set of -tuples. For example, multiplication of
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s is a function of two variables, or bivariate function, whose domain is the set of all pairs (2-tuples) of integers, and whose codomain is the set of integers. The same is true for every
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
. More generally, every
mathematical operation In mathematics, an operation is a Function (mathematics), function which takes zero or more input values (also called "''operands''" or "arguments") to a well-defined output value. The number of operands is the arity of the operation. The most c ...
is defined as a multivariate function. The
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
X_1\times\cdots\times X_n of sets X_1, \ldots, X_n is the set of all -tuples (x_1, \ldots, x_n) such that x_i\in X_i for every with 1 \leq i \leq n. Therefore, a function of variables is a function :f\colon U\to Y, where the domain has the form :U\subseteq X_1\times\cdots\times X_n. When using function notation, one usually omits the parentheses surrounding tuples, writing f(x_1,x_2) instead of f((x_1,x_2)). In the case where all the X_i are equal to the set \R of
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, one has a
function of several real variables In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function o ...
. If the X_i are equal to the set \C of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, one has a
function of several complex variables The theory of functions of several complex variables is the branch of mathematics dealing with complex-valued functions. The name of the field dealing with the properties of function of several complex variables is called several complex variable ...
. It is common to also consider functions whose codomain is a product of sets. For example,
Euclidean division In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
maps every pair of integers with to a pair of integers called the ''quotient'' and the ''remainder'': :\begin \text\colon\quad \Z\times (\Z\setminus \) &\to \Z\times\Z\\ (a,b) &\mapsto (\operatorname(a,b),\operatorname(a,b)). \end The codomain may also be a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
. In this case, one talks of a
vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
. If the domain is contained in a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, or more generally a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, a vector-valued function is often called a vector field.


In calculus

The idea of function, starting in the 17th century, was fundamental to the new
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. At that time, only
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
functions of a real variable were considered, and all functions were assumed to be
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
. But the definition was soon extended to functions of several variables and to
functions of a complex variable Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
. In the second half of the 19th century, the mathematically rigorous definition of a function was introduced, and functions with arbitrary domains and codomains were defined. Functions are now used throughout all areas of mathematics. In introductory
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
, when the word ''function'' is used without qualification, it means a real-valued function of a single real variable. The more general definition of a function is usually introduced to second or third year college students with
STEM Stem or STEM may refer to: Plant structures * Plant stem, a plant's aboveground axis, made of vascular tissue, off which leaves and flowers hang * Stipe (botany), a stalk to support some other structure * Stipe (mycology), the stem of a mushro ...
majors, and in their senior year they are introduced to calculus in a larger, more rigorous setting in courses such as
real analysis In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include converg ...
and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
.


Real function

A ''real function'' is a
real-valued In mathematics, value may refer to several, strongly related notions. In general, a mathematical value may be any definite mathematical object. In elementary mathematics, this is most often a number – for example, a real number such as or an i ...
function of a real variable In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function (mathematics), function whose domain of a function, domain is the real numbers \mathbb, or ...
, that is, a function whose codomain is the field of real numbers and whose domain is a set of
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 that contains an interval. In this section, these functions are simply called ''functions''. The functions that are most commonly considered in mathematics and its applications have some regularity, that is they are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
,
differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
, and even analytic. This regularity insures that these functions can be visualized by their
graphs Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
. In this section, all functions are differentiable in some interval. Functions enjoy
pointwise operation In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined ...
s, that is, if and are functions, their sum, difference and product are functions defined by :\begin (f+g)(x)&=f(x)+g(x)\\ (f-g)(x)&=f(x)-g(x)\\ (f\cdot g)(x)&=f(x)\cdot g(x)\\ \end. The domains of the resulting functions are the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of the domains of and . The quotient of two functions is defined similarly by :\frac fg(x)=\frac, but the domain of the resulting function is obtained by removing the zeros of from the intersection of the domains of and . The
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtrac ...
s are defined by
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s, and their domain is the whole set of real numbers. They include
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properties ...
s,
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
s and
quadratic function In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before 20th century, the distinction was unclear between a polynomial ...
s.
Rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s are quotients of two polynomial functions, and their domain is the real numbers with a finite number of them removed to avoid
division by zero In mathematics, division by zero is division (mathematics), division where the divisor (denominator) is 0, zero. Such a division can be formally expression (mathematics), expressed as \tfrac, where is the dividend (numerator). In ordinary ari ...
. The simplest rational function is the function x\mapsto \frac 1x, whose graph is a
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, cal ...
, and whose domain is the whole
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
except for 0. The
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 ...
of a real differentiable function is a real function. An
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
of a continuous real function is a real function that has the original function as a derivative. For example, the function x\mapsto\frac 1x is continuous, and even differentiable, on the positive real numbers. Thus one antiderivative, which takes the value zero for , is a differentiable function called the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
. A real function is
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...
in an interval if the sign of \frac does not depend of the choice of and in the interval. If the function is differentiable in the interval, it is monotonic if the sign of the derivative is constant in the interval. If a real function is monotonic in an interval , it has 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 ...
, which is a real function with domain and image . This is how
inverse trigonometric functions In mathematics, the inverse trigonometric functions (occasionally also called arcus functions, antitrigonometric functions or cyclometric functions) are the inverse functions of the trigonometric functions (with suitably restricted Domain of a fu ...
are defined in terms of
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 ...
, where the trigonometric functions are monotonic. Another example: the natural logarithm is monotonic on the positive real numbers, and its image is the whole real line; therefore it has an inverse function that is a
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 ...
between the real numbers and the positive real numbers. This inverse is the
exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...
. Many other real functions are defined either by the implicit function theorem (the inverse function is a particular instance) or as solutions of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. For example, the
sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is oppo ...
and the cosine functions are the solutions of the
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b( ...
:y''+y=0 such that :\sin 0=0, \quad \cos 0=1, \quad\frac(0)=1, \quad\frac(0)=0.


Vector-valued function

When the elements of the codomain of a function are vectors, the function is said to be a vector-valued function. These functions are particularly useful in applications, for example modeling physical properties. For example, the function that associates to each point of a fluid its
velocity vector Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
is a vector-valued function. Some vector-valued functions are defined on a subset of \mathbb^n or other spaces that share geometric or
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
properties of \mathbb^n, such as
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Ne ...
. These vector-valued functions are given the name ''vector fields''.


Function space

In
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, and more specifically in
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
, a function space is a set of scalar-valued or
vector-valued function A vector-valued function, also referred to as a vector function, is a mathematical function of one or more variables whose range is a set of multidimensional vectors or infinite-dimensional vectors. The input of a vector-valued function could ...
s, which share a specific property and form a
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
. For example, the real
smooth function In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s with a
compact support In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest ...
(that is, they are zero outside some
compact set In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
) form a function space that is at the basis of the theory of distributions. Function spaces play a fundamental role in advanced mathematical analysis, by allowing the use of their algebraic and
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...
properties for studying properties of functions. For example, all theorems of existence and uniqueness of solutions of ordinary or
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
s result of the study of function spaces.


Multi-valued functions

Several methods for specifying functions of real or complex variables start from a local definition of the function at a point or on a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of a point, and then extend by continuity the function to a much larger domain. Frequently, for a starting point x_0, there are several possible starting values for the function. For example, in defining the
square root In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or  ⋅ ) is . For example, 4 and −4 are square roots of 16, because . E ...
as the inverse function of the square function, for any positive real number x_0, there are two choices for the value of the square root, one of which is positive and denoted \sqrt , and another which is negative and denoted -\sqrt . These choices define two continuous functions, both having the nonnegative real numbers as a domain, and having either the nonnegative or the nonpositive real numbers as images. When looking at the graphs of these functions, one can see that, together, they form a single
smooth curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that a ...
. It is therefore often useful to consider these two square root functions as a single function that has two values for positive , one value for 0 and no value for negative . In the preceding example, one choice, the positive square root, is more natural than the other. This is not the case in general. For example, let consider the
implicit function In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit functi ...
that maps to a
root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
of x^3-3x-y =0 (see the figure on the right). For one may choose either 0, \sqrt 3,\text -\sqrt 3 for . By the implicit function theorem, each choice defines a function; for the first one, the (maximal) domain is the interval and the image is ; for the second one, the domain is and the image is ; for the last one, the domain is and the image is . As the three graphs together form a smooth curve, and there is no reason for preferring one choice, these three functions are often considered as a single ''multi-valued function'' of that has three values for , and only one value for and . Usefulness of the concept of multi-valued functions is clearer when considering complex functions, typically
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex an ...
s. The domain to which a complex function may be extended by
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
generally consists of almost the whole
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
. However, when extending the domain through two different paths, one often gets different values. For example, when extending the domain of the square root function, along a path of complex numbers with positive imaginary parts, one gets for the square root of −1; while, when extending through complex numbers with negative imaginary parts, one gets . There are generally two ways of solving the problem. One may define a function that is not
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
along some curve, called a
branch cut In the mathematical field of complex analysis, a branch point of a multi-valued function (usually referred to as a "multifunction" in the context of complex analysis) is a point such that if the function is n-valued (has n values) at that point, a ...
. Such a function is called the
principal value In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. The simplest case arises in taking the square root of a positive ...
of the function. The other way is to consider that one has a ''multi-valued function'', which is analytic everywhere except for isolated singularities, but whose value may "jump" if one follows a closed loop around a singularity. This jump is called the
monodromy In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ...
.


In the foundations of mathematics and set theory

The definition of a function that is given in this article requires the concept of
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, since the domain and the codomain of a function must be a set. This is not a problem in usual mathematics, as it is generally not difficult to consider only functions whose domain and codomain are sets, which are well defined, even if the domain is not explicitly defined. However, it is sometimes useful to consider more general functions. For example, the
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, the ...
may be considered as a function x\mapsto \. Its domain would include all sets, and therefore would not be a set. In usual mathematics, one avoids this kind of problem by specifying a domain, which means that one has many singleton functions. However, when establishing foundations of mathematics, one may have to use functions whose domain, codomain or both are not specified, and some authors, often logicians, give precise definition for these weakly specified functions.; ; These generalized functions may be critical in the development of a formalization of the
foundations of mathematics Foundations of mathematics is the study of the philosophy, philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the natu ...
. For example,
Von Neumann–Bernays–Gödel set theory In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collect ...
, is an extension of the set theory in which the collection of all sets is a
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...
. This theory includes the
replacement axiom In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinit ...
, which may be stated as: If is a set and is a function, then is a set.


In computer science

In
computer programming Computer programming is the process of performing a particular computation (or more generally, accomplishing a specific computing result), usually by designing and building an executable computer program. Programming involves tasks such as ana ...
, a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
is, in general, a piece of a
computer program A computer program is a sequence or set of instructions in a programming language for a computer to execute. Computer programs are one component of software, which also includes documentation and other intangible components. A computer program ...
, which implements the abstract concept of function. That is, it is a program unit that produces an output for each input. However, in many
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 every
subroutine In computer programming, a function or subroutine is a sequence of program instructions that performs a specific task, packaged as a unit. This unit can then be used in programs wherever that particular task should be performed. Functions may ...
is called a function, even when there is no output, and when the functionality consists simply of modifying some data in the
computer memory In computing, memory is a device or system that is used to store information for immediate use in a computer or related computer hardware and digital electronic devices. The term ''memory'' is often synonymous with the term ''primary storage ...
.
Functional programming In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declar ...
is the
programming paradigm Programming paradigms are a way to classify programming languages based on their features. Languages can be classified into multiple paradigms. Some paradigms are concerned mainly with implications for the execution model of the language, suc ...
consisting of building programs by using only subroutines that behave like mathematical functions. For example, if_then_else is a function that takes three functions as arguments, and, depending on the result of the first function (''true'' or ''false''), returns the result of either the second or the third function. An important advantage of functional programming is that it makes easier
program proof In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of intended algorithms underlying a system with respect to a certain formal specification or property, using formal method ...
s, as being based on a well founded theory, the
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 ...
(see below). Except for computer-language terminology, "function" has the usual mathematical meaning in
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
. In this area, a property of major interest is the
computability Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory within mathematical logic and the theory of computation within computer science. The computability of a problem is close ...
of a function. For giving a precise meaning to this concept, and to the related concept of
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
, several
models of computation In computer science, and more specifically in computability theory and computational complexity theory, a model of computation is a model which describes how an output of a mathematical function is computed given an input. A model describes how ...
have been introduced, the old ones being
general recursive function In mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as i ...
s,
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 ...
and
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
. The fundamental theorem of
computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since e ...
is that these three models of computation define the same set of computable functions, and that all the other models of computation that have ever been proposed define the same set of computable functions or a smaller one. The
Church–Turing thesis In computability theory, the Church–Turing thesis (also known as computability thesis, the Turing–Church thesis, the Church–Turing conjecture, Church's thesis, Church's conjecture, and Turing's thesis) is a thesis about the nature of comp ...
is the claim that every philosophically acceptable definition of a ''computable function'' defines also the same functions. General recursive functions are
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 from integers to integers that can be defined from *
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properties ...
s, *
successor Successor may refer to: * An entity that comes after another (see Succession (disambiguation)) Film and TV * ''The Successor'' (film), a 1996 film including Laura Girling * ''The Successor'' (TV program), a 2007 Israeli television program Mus ...
, and *
projection Projection, projections or projective may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphic ...
functions via the operators *
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 ...
, *
primitive recursion 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 ...
, and * minimization. Although defined only for functions from integers to integers, they can model any computable function as a consequence of the following properties: * a computation is the manipulation of finite sequences of symbols (digits of numbers, formulas, ...), * every sequence of symbols may be coded as a sequence of
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s, * a bit sequence can be interpreted as the
binary representation A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one). The base-2 numeral system is a positional notation ...
of an integer.
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 ...
is a theory that defines computable functions without using
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
, and is the theoretical background of functional programming. It consists of ''terms'' that are either variables, function definitions ('-terms), or applications of functions to terms. Terms are manipulated through some rules, (the -equivalence, the -reduction, and the -conversion), which are the
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or f ...
s of the theory and may be interpreted as rules of computation. In its original form, lambda calculus does not include the concepts of domain and codomain of a function. Roughly speaking, they have been introduced in the theory under the name of ''type'' in
typed lambda calculus A typed lambda calculus is a typed formalism that uses the lambda-symbol (\lambda) to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a ...
. Most kinds of typed lambda calculi can define fewer functions than untyped lambda calculus.


See also


Subpages

*
List of types of functions Functions can be identified according to the properties they have. These properties describe the functions' behaviour under certain conditions. A parabola is a specific type of function. Relative to set theory These properties concern the domain ...
*
List of functions In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed ...
*
Function fitting Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is ...
*
Implicit function In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit functi ...


Generalizations

*
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 ...
*
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" ...
*
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 ...
*
Microfunction Algebraic analysis is an area of mathematics that deals with systems of Partial differential equation, linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of Function (mathematic ...
*
Distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations * Probability distribution, the probability of a particular value or value range of a vari ...
*
Functor In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...


Related topics

*
Associative array In computer science, an associative array, map, symbol table, or dictionary is an abstract data type that stores a collection of (key, value) pairs, such that each possible key appears at most once in the collection. In mathematical terms an as ...
*
Closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
*
Elementary function In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponen ...
*
Functional 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 sy ...
*
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 predicate In formal logic and related branches of mathematics, a functional predicate, or function symbol, is a logical symbol that may be applied to an object term to produce another object term. Functional predicates are also sometimes called mappings, but ...
*
Functional programming In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declar ...
*
Parametric equation In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric obj ...
*
Set function In mathematics, especially measure theory, a set function is a function whose domain is a family of subsets of some given set and that (usually) takes its values in the extended real number line \R \cup \, which consists of the real numbers \R a ...
*
Simple function In the mathematical field of real analysis, a simple function is a real (or complex)-valued function over a subset of the real line, similar to a step function. Simple functions are sufficiently "nice" that using them makes mathematical reasoning, ...


Notes


References


Sources

* * * * * * *


Further reading

* * * * * * * * An approachable and diverting historical presentation. * * Reichenbach, Hans (1947) ''Elements of Symbolic Logic'', Dover Publishing Inc., New York, . * *


External links

*
The Wolfram Functions Site
gives formulae and visualizations of many mathematical functions.
NIST Digital Library of Mathematical Functions
{{Authority control Basic concepts in set theory Elementary mathematics