In

mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, the domain of a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

is the set of inputs accepted by the function. It is sometimes denoted by $\backslash operatorname(f)$, where is the function.
More precisely, given a function $f\backslash colon\; X\backslash to\; Y$, the domain of is . Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it.
In the special case that and are both subsets of $\backslash R$, the function can be graphed in the Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of number, numerical coordinates, which are the positive and negative numbers, signed distances ...

. In this case, the domain is represented on the -axis of the graph, as the projection of the graph of the function onto the -axis.
For a function $f\backslash colon\; X\backslash to\; Y$, the set is called the codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, and the set of values attained by the function (which is a subset of ) is called its 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 ...

or image
File:TEIDE.JPG, An Synthetic aperture radar, SAR radar imaging, radar image acquired by the SIR-C/X-SAR radar on board the Space Shuttle Endeavour shows the Teide volcano. The city of Santa Cruz de Tenerife is visible as the purple and white a ...

.
Any function can be restricted to a subset of its domain. The restriction of $f\; \backslash colon\; X\; \backslash to\; Y$ to $A$, where $A\backslash subseteq\; X$, is written as $\backslash left.\; f\; \backslash \_A\; \backslash colon\; A\; \backslash to\; Y$.
Natural domain

If areal function
In mathematical analysis, and applications in geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. ...

is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

, and the set of real numbers on which the formula can be evaluated to a real number is called the natural domain or domain of definition of . In many contexts, a partial function is called simply a ''function'', and its natural domain is called simply its ''domain''.
Examples

* The function $f$ defined by $f(x)=\backslash frac$ cannot be evaluated at 0. Therefore the natural domain of is $\backslash mathbb\; \backslash setminus\; \backslash $. * In contrast, if $f$ is thepiecewise
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

function $f(x)\; =\; \backslash begin\; 1/x\&x\backslash not=0\backslash \backslash \; 0\&x=0\; \backslash end,$ then ''$f$'' is defined for all real numbers, and its natural domain is $\backslash mathbb$.
* The function $x\backslash mapsto\backslash sqrt\; x$ has as its natural domain the non-negative real numbers, which can be denoted by $\backslash mathbb\; R\_$, by the interval $(0,\backslash infty)$, or by $\backslash $.
* The tangent function
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$\backslash tan\; x$ has as its natural domain the set of all real numbers which are not of the form $\backslash tfrac\; +\; k\; \backslash pi,$ where is any integer
An integer (from the Latin
Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

.
Other uses

The word "domain" is used with other related meanings in some areas of mathematics. Intopology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

, a domain is a connected open set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

. In real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

and complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions
Function or functionality may refer to:
Computing
* Function key
A function key i ...

, a domain is an open
Open or OPEN may refer to:
Music
* Open (band)
Open is a band.
Background
Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...

connected subset of a real
Real may refer to:
* Reality
Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ...

or complex
The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London
, mottoeng = Let all come who by merit deserve the most reward
, established =
, type = Public university, Public rese ...

vector space. In the study of partial differential equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s, a domain is the open connected subset of the Euclidean space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensi ...

$\backslash mathbb^$ where a problem is posed (i.e., where the unknown function(s) are defined).
Set theoretical notions

For example, it is sometimes convenient inset theory
illustrating the intersection of two sets
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 ...

to permit the domain of a function to be a proper class
Proper may refer to:
Mathematics
* Proper map
In mathematics, a function (mathematics), function between topological spaces is called proper if inverse images of compact space, compact subsets are compact. In algebraic geometry, the analogous ...

, in which case there is formally no such thing as a triple . With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form ., p. 91 ( quote 1 quote 2; , p. 8P. is an abbreviation or acronym that may refer to:
* Page (paper)
A page is one side of a leaf
A leaf (plural leaves) is the principal lateral appendage of the vascular plant plant stem, stem, usually borne above ground and specialized f ...

Mac Lane, in , p. 232P. is an abbreviation or acronym that may refer to:
* Page (paper)
A page is one side of a leaf
A leaf (plural leaves) is the principal lateral appendage of the vascular plant plant stem, stem, usually borne above ground and specialized f ...

, p. 91 , p. 89/ref>
See also

* Attribute domain *Bijection, injection and surjection
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

* Codomain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

* Domain decomposition
In mathematics, numerical analysis, and numerical partial differential equations, domain decomposition methods solve a boundary value problem by splitting it into smaller boundary value problems on subdomains and iterating to coordinate the solutio ...

* Effective domain
* Image (mathematics)
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

* Lipschitz domainIn mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a Domain (mathematical analysis), domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lips ...

* Naive set theory
Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sens ...

* Support (mathematics)
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Notes

References

* {{Mathematical logic Functions and mappingsBasic concepts in set theory{{Commons
This category is for the foundational concepts of naive set theory, in terms of which contemporary mathematics is typically expressed.
Mathematical concepts ...