In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a function from a
set to a set assigns to each element of exactly one element of .
[; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously.] The set is called the
domain of the function and the set is called the
codomain
In mathematics, a codomain, counter-domain, or set of destination of a function is a 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 ...
of the function.
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, Hydrostatic equilibrium, rounded Astronomical object, astronomical body that is generally required to be in orbit around a star, stellar remnant, or brown dwarf, and is not one itself. The Solar System has eight planets b ...
is a ''function'' of time.
Historically, the concept was elaborated with the
infinitesimal calculus
Calculus 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 arithmetic operations.
Originally called infinitesimal calculus or "the calculus of ...
at the end of the 17th century, and, until the 19th century, the functions that were considered were
differentiable (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 Set (mathematics), 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 mathema ...
, and this greatly increased the possible applications of the concept.
A function is often denoted by a letter such as , or . The value of a function at an element of its domain (that is, the element of the codomain that is associated with ) is denoted by ; for example, the value of at is denoted by . Commonly, a specific function is defined by means of an
expression depending on , such as
in this case, some computation, called , may be needed for deducing the value of the function at a particular value; for example, if
then
Given its domain and its codomain, a function is uniquely represented by the set of all
pairs , 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
In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
of a point in the plane.
Functions are widely used in
science
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
,
engineering
Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.
The concept of a function has evolved significantly over centuries, from its informal origins in ancient mathematics to its formalization in the 19th century. See
History of the function concept for details.
Definition

A function from a
set to a set is an assignment of one element of to each element of . The set is called the
domain of the function and the set is called the
codomain
In mathematics, a codomain, counter-domain, or set of destination of a function is a 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 ...
of the function.
If the element in is assigned to in by the function , one says that ''maps'' to , and this is commonly written
In this notation, is the ''
argument
An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persu ...
'' or ''
variable'' of the function.
A specific element of is a ''value of the variable'', and the corresponding element of is the ''value of the function'' at , or the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of under the function. The ''image of a function'', sometimes called its
range, is the set of the images of all elements in the domain.
A function , its domain , and its codomain are often specified by the notation
One may write
instead of
, where the symbol
(read '
maps to') is used to specify where a particular element in the domain is mapped to by . This allows the definition of a function without naming. For example, the
square function is the function
The domain and codomain are not always explicitly given when a function is defined. In particular, it is common that one might only know, without some (possibly difficult) computation, that the domain of a specific function is contained in a larger set. For example, if
is a
real function
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
, the determination of the domain of the function
requires knowing the
zeros of This is one of the reasons for which, 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 ( ...
, "a function may refer to a function having a proper subset of as a domain.
[The true domain of such a function is often called the ''domain of definition'' of the function.] For example, a "function from the reals to the reals" may refer to a
real-valued function of a
real variable whose domain is a proper subset 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, typically a subset that contains a non-empty
open interval
In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
. 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 the whole itself) to . The subset , that is, the '' domain'' of viewed as a function, is called the domain of definition or natural domain ...
.
A function on a set means a function from the domain , without specifying a codomain. However, some authors use it as shorthand for saying that the function is .
Formal definition
Diagram of a function
file:Injection keine Injektion 1.svg, Diagram of a relation that is not a function. One reason is that 2 is the first element in more than one ordered pair. Another reason is that neither 3 nor 4 are the first element (input) of any ordered pair.
The above definition of a function is essentially that of the founders of
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, Leibniz, Isaac Newton, Newton and Euler. However, it cannot be formal proof, formalized, since there is no mathematical definition of an "assignment". It is only at the end of the 19th century that the first formal definition of a function could be provided, in terms of
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), 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 mathema ...
. This set-theoretic definition is based on the fact that a function establishes a ''relation'' between the elements of the domain and some (possibly all) elements of the codomain. Mathematically, a
binary relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
between two sets and is a
subset
In mathematics, a 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 a ...
of the set of all
ordered pair
In mathematics, an ordered pair, denoted (''a'', ''b''), is a pair of objects in which their order is significant. The ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a''), unless ''a'' = ''b''. In contrast, the '' unord ...
s
such that
and
The set of all these pairs is called the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of and and denoted
Thus, the above definition may be formalized as follows.
A ''function'' with domain and codomain is a binary relation between and that satisfies the two following conditions:
* For every
in
there exists
in
such that
* If
and
then
This definition may be rewritten more formally, without referring explicitly to the concept of a relation, but using more notation (including
set-builder notation):
A function is formed by three sets, the ''domain''
the ''codomain''
and the ''graph''
that satisfy the three following conditions.
*
*
*
Partial functions
Partial functions are defined similarly to ordinary functions, with the "total" condition removed. That is, a ''partial function'' from to is a binary relation between and such that, for every
there is ''at most one'' in such that
Using functional notation, this means that, given
either
is in , or it is undefined.
The set of the elements of such that
is defined and belongs to is called the ''domain of definition'' of the function. A partial function from to is thus an ordinary function that has as its domain a subset of called the domain of definition of the function. If the domain of definition equals , one often says that the partial function is a ''total function''.
In several areas of mathematics, the term "function" refers to partial functions rather than to ordinary (total) functions. This is typically the case when functions may be specified in a way that makes difficult or even impossible to determine their domain.
In
calculus
Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations.
Originally called infinitesimal calculus or "the ...
, a ''real-valued function of a real variable'' or ''
real function
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers \mathbb, or a subset of \mathbb that contains an inter ...
'' is a partial function from the 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s to itself. Given a real function
its
multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a ra ...
is also a real function. The determination of the domain of definition of a multiplicative inverse of a (partial) function amounts to compute the
zeros of the function, the values where the function is defined but not its multiplicative inverse.
Similarly, a ''
function of a complex variable'' is generally a partial function whose domain of definition is 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 for ...
s
. The difficulty of determining the domain of definition of a
complex function is illustrated by the multiplicative inverse of the
Riemann zeta function: the determination of the domain of definition of the function
is more or less equivalent to the proof or disproof of one of the major open problems in mathematics, the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
.
In
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 ex ...
, a
general recursive function is a partial function from the integers to the integers whose values can be computed by an
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
(roughly speaking). The domain of definition of such a function is the set of inputs for which the algorithm does not run forever. A fundamental theorem of computability theory is that there cannot exist an algorithm that takes an arbitrary general recursive function as input and tests whether belongs to its domain of definition (see
Halting problem
In computability theory (computer science), computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run for ...
).
Multivariate functions

A multivariate function, multivariable 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.
Formally, a function of variables is a function whose domain is a set of -tuples.
[ may also be 1, thus subsuming functions as defined above. For , each constant is a special case of a multivariate function, too.] For example, multiplication of
integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s is a function of two variables, or bivariate function, whose domain is the set of all
ordered 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, a binary operation ...
. The graph of a bivariate surface over a two-dimensional real domain may be interpreted as defining a
parametric surface, as used in, e.g.,
bivariate interpolation.
Commonly, an -tuple is denoted enclosed between parentheses, such as in
When using
functional notation, one usually omits the parentheses surrounding tuples, writing
instead of
Given sets
the set of all -tuples
such that
is called the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is
A\times B = \.
A table c ...
of
and denoted
Therefore, a multivariate function is a function that has a Cartesian product or a
proper subset of a Cartesian product as a domain.
where the domain has the form
If all the
are equal to the 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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s or to the set
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 for ...
s, one talks respectively of a
function of several real variables or of a
function of several complex variables.
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
The functional notation requires that a name is given to the function, which, in the case of a unspecified function is often the letter . Then, the application of the function to an argument is denoted by its name followed by its argument (or, in the case of a multivariate functions, its arguments) enclosed between parentheses, such as in
The argument between the parentheses may be a
variable, often , that represents an arbitrary element of the domain of the function, a specific element of the domain ( in the above example), or an
expression that can be evaluated to an element of the domain (
in the above example). The use of a unspecified variable between parentheses is useful for defining a function explicitly such as in "let
".
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 polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
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 is customarily used instead, such as "" for the
sine function, in contrast to italic font for single-letter symbols.
The functional notation is often used colloquially for referring to a function and simultaneously naming its argument, such as in "let
be a function". This is an
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 ...
that is useful for a simpler formulation.
Arrow notation
Arrow notation defines the rule of a function inline, without requiring a name to be given to the function. It uses the ↦ arrow symbol, pronounced "
maps to". For example,
is the function which takes a real number as input and outputs that number plus 1. Again, a domain and codomain of
is implied.
The domain and codomain can also be explicitly stated, for example:
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
is a function in two variables, and we want to refer to a
partially applied function produced by fixing the second argument to the value without introducing a new function name. The map in question could be denoted
using the arrow notation. The expression
(read: "the map taking to of comma nought") represents this new function with just one argument, whereas the expression refers to the value of the function at the
Index notation
Index notation may be used instead of functional notation. That is, instead of writing , one writes
This is typically the case for functions whose domain is the set of the
natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
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 cal ...
, and, in this case the element
is called the th element of the sequence.
The index notation can also be 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
(see above) would be denoted
using index notation, if we define the collection of maps
by the formula
for all
.
Dot notation
In the notation
the symbol does not represent any value; it is simply a
placeholder, 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, centered dot or centred dot, is a punctuation mark consisting of a vertically centered dot used for interword separation in Classical Latin. ( Word-separating spaces did not appe ...
"". This may be useful for distinguishing the function from its value at .
For example,
may stand for the function
, and
may stand for a function defined by an
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
with variable upper bound:
.
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 matrix (mathemat ...
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 (for example, Inner product space#Definition, inner product, Norm (mathematics ...
,
linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field (mat ...
s and the
vectors they act upon are denoted using a
dual pair to show the underlying
duality. This is similar to the use of
bra–ket notation
Bra–ket notation, also called Dirac notation, is a notation for linear algebra and linear operators on complex vector spaces together with their dual space both in the finite-dimensional and infinite-dimensional case. It is specifically de ...
in quantum mechanics. In
logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the study of deductively valid inferences or logical truths. It examines how conclusions follow from premises based on the structure o ...
and the
theory of computation, the function notation of
lambda calculus
In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using var ...
is used to explicitly express the basic notions of function
abstraction
Abstraction is a process where general rules and concepts are derived from the use and classifying of specific examples, literal (reality, real or Abstract and concrete, concrete) signifiers, first principles, or other methods.
"An abstraction" ...
and
application. In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
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 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.
Functions of more than one variable
In some cases the argument of a function may be an ordered pair of elements taken from some set or sets. For example, a function can be defined as mapping any pair of real numbers
to the sum of their squares,
. Such a function is commonly written as
and referred to as "a function of two variables". Likewise one can have a function of three or more variables, with notations such as
,
.
Other terms
A function may also be 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 particular ''map'' may be 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 mapping V \to W between two vector spaces that p ...
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)
whe ...
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,
[ use "function" only to refer to maps for which the ]codomain
In mathematics, a codomain, counter-domain, or set of destination of a function is a 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 ...
is a subset of the real 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, such as in a parametric curve. Examples include the mathematical models ...
s, a map denotes an evolution function used to create discrete dynamical systems. See also Poincaré map.
Whichever definition of ''map'' is used, related terms like '' domain'', ''codomain
In mathematics, a codomain, counter-domain, or set of destination of a function is a 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 ...
'', ''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 of its codomain; that is, implies (equivalently by contraposition, impl ...
'', '' continuous'' have the same meaning as for a function.
Specifying a function
Given a function , by definition, to each element of the domain of the function , there is a unique element associated to it, the value of at . There are several ways to specify or describe how is related to , 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 ...
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 .
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 , then one can define a function by
By a formula
Functions are often defined by an expression that describes a combination of arithmetic operations
Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and Division (mathematics), division. In a wider sense, it also includes exponentiation, extraction of nth root, ...
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, can be defined by the formula , for .
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 y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
s occur in the definition of a function from to 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, defines a function whose domain is because is always positive if is a real number. On the other hand, 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 can be classified by the nature of formulas that define them:
* A quadratic function
In mathematics, a quadratic function of a single variable (mathematics), variable is a function (mathematics), function of the form
:f(x)=ax^2+bx+c,\quad a \ne 0,
where is its variable, and , , and are coefficients. The mathematical expression, e ...
is a function that may be written where are constants.
* More generally, a polynomial function is a function that can be defined by a formula involving only additions, subtractions, multiplications, and exponentiation
In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
to nonnegative integer powers. For example, and are polynomial functions of .
* 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 ...
is the same, with divisions also allowed, such as and
* An algebraic function 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, a ...
[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 of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
s and exponential functions allowed.
Inverse and implicit functions
A function with domain and codomain , is bijective
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
, 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 ...
of is the function that maps to the element such that . For example, the natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
is a bijective function from the positive real numbers to the real numbers. It thus has an inverse, called the exponential function, that maps the real numbers onto the positive numbers.
If a function is not bijective, it may occur that one can select subsets and such that the restriction of to is a bijection from to , and has thus an inverse. The inverse trigonometric functions are defined this way. For example, the cosine function induces, by restriction, a bijection from the interval onto the interval , and its inverse function, called arccosine, maps onto . The other inverse trigonometric functions are defined similarly.
More generally, given a binary relation
In mathematics, a binary relation associates some elements of one Set (mathematics), set called the ''domain'' with some elements of another set called the ''codomain''. Precisely, a binary relation over sets X and Y is a set of ordered pairs ...
between two sets and , let be a subset of such that, for every there is some such that . If one has a criterion allowing selecting such a for every this defines a function called an implicit function, 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 Eucli ...
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 but, in more complicated examples, this is impossible. For example, the relation defines as an implicit function of , called the Bring radical, which has 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 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 of another function. This is the case of the natural logarithm
The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...
, which is the antiderivative of that is 0 for . Another common example is the error function.
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 equations. The simplest example is probably the exponential function, which can be defined as the unique function that is equal to its derivative and takes the value 1 for .
Power series can be used to define functions on the domain in which they converge. For example, the exponential function is given by . 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 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 for ...
s, the disc of convergence 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 ne ...
allows enlarging further the domain for including almost the whole complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. This process is the method that is generally used for defining the logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , the ...
, the exponential 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 cal ...
s, are sometimes 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 (mathematics), 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 ...
function on the nonnegative integers () is a basic example, as it can be defined by the recurrence relation
and the initial condition
Representing a function
A graph 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 charts.
Graphs and plots
Given a function its ''graph'' is, formally, the set
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 duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s (or may be identified with such subsets, e.g. intervals), an element may be identified with a point having coordinates in a 2-dimensional coordinate system, e.g. the Cartesian plane. Parts of this may create a 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
consisting of all points with coordinates for yields, when depicted in Cartesian coordinates, the well known parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
. If the same quadratic function with the same formal graph, consisting of pairs of numbers, is plotted instead in polar coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are
*the point's distance from a reference ...
the plot obtained is Fermat's spiral.
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 defined as can be represented by the familiar multiplication table
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 mathematics, 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 ...
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
A bar chart can represent a function whose domain is a finite set, the natural number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s, or the integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
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 or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
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 is not equal to if and only if , although their graphs are both the empty set
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
.
* For every set and every singleton set , 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 the ''canonical surjection'' of onto its image is the function from to that maps to .
* For every subset
In mathematics, a 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 a ...
of a set , the inclusion map of into is the injective (see below) function that maps every element of to itself.
* The identity function on a set , often denoted by , is the inclusion of into itself.
Function composition
Given two functions and such that the domain of is the codomain of , their ''composition'' is the function defined by
That is, the value of is obtained by first applying to to obtain and then applying to the result to obtain . In this notation, the function that is applied first is always written on the right.
The composition is an operation on functions that is defined only if the codomain of the first function is the domain of the second one. Even when both and 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. Perhaps most familiar as a pr ...
, that is, the functions and need not be equal, but may deliver different values for the same argument. For example, let and , then and agree just for
The function composition is associative
In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
in the sense that, if one of and is defined, then the other is also defined, and they are equal, that is, Therefore, it is usual to just write
The identity functions and are respectively a right identity and a left identity for functions from to . That is, if is a function with domain , and codomain , one has
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 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,
The ''image'' of is the image of the whole domain, that is, . It is also called the range 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'' or '' preimage'' 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 and is given by the equation
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 and is given by the equation
For example, the preimage of under the square function 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 of an element of the codomain may be empty or contain any number of elements. For example, if is the function from the integers to themselves that maps every integer to 0, then .
If is a function, and are subsets of , and and are subsets of , then one has the following properties:
*
*
*
*
*
*
The preimage by of an element of the codomain is sometimes called, in some contexts, the fiber
Fiber (spelled fibre in British English; from ) 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 inco ...
of under .
If a function has an inverse (see below), this inverse is denoted In this case may denote either the image by or the preimage by of . This is not a problem, as these sets are equal. The notation and 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