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In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by $\operatorname\left(f\right)$ or $\operatornamef$, where is the function. More precisely, given a function $f\colon X\to Y$, 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 $\R$, the function can be graphed in the
Cartesian coordinate system 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 i ...
. 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 $f\colon X\to Y$, 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 ...
, and the set of values attained by the function (which is a subset of ) is called its range or image. Any function can be restricted to a subset of its domain. The restriction of $f \colon X \to Y$ to $A$, where $A\subseteq X$, is written as $\left. f \_A \colon A \to Y$.

# Natural domain

If 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 interv ...
is given by a formula, it may be not defined for some values of the variable. In this case, it is a partial function, 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 $f\left(x\right)=\frac$ cannot be evaluated at 0. Therefore the natural domain of $f$ is the set of real numbers excluding 0, which can be denoted by $\mathbb \setminus \$ or $\$. * The piecewise function $f$ defined by $f\left(x\right) = \begin 1/x&x\not=0\\ 0&x=0 \end,$ has as its natural domain the set $\mathbb$ of real numbers. * 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 ...
function $f\left(x\right)=\sqrt x$ has as its natural domain the set of non-negative real numbers, which can be denoted by $\mathbb R_$, the interval , or $\$. * The tangent function, denoted $\tan$, has as its natural domain the set of all real numbers which are not of the form $\tfrac + k \pi$ for some integer $k$, which can be written as $\mathbb R \setminus \$.

# Other uses

The word "domain" is used with other related meanings in some areas of mathematics. In
topology 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 ...
, a domain is a connected open set. In
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ... and complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ... , a domain is an open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (Y ... connected subset of a real Real may refer to: Currencies * Brazilian real (R$) * 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 vector space. In the study of partial differential equations, a domain is the open connected subset of the
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
$\mathbb^$ where a problem is posed (i.e., where the unknown function(s) are defined).

# Set theoretical notions

For example, it is sometimes convenient in
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 concer ...
to permit the domain of a function to be a
proper class Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
, 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 Quote is a hypernym of quotation, as the repetition or copy of a prior statement or thought. Quotation marks are punctuation marks that indicate a quotation. Both ''quotation'' and ''quotation marks'' are sometimes abbreviated as "quote(s)". Co ...
; ,
p. 8 P. is an abbreviation or acronym that may refer to: * Page (paper), where the abbreviation comes from Latin ''pagina'' * Paris Herbarium, at the ''Muséum national d'histoire naturelle'' * ''Pani'' (Polish), translating as Mrs. * The ''Pacific Rep ...
Mac Lane, in ,
p. 232 P. is an abbreviation or acronym that may refer to: * Page (paper), where the abbreviation comes from Latin ''pagina'' * Paris Herbarium, at the ''Muséum national d'histoire naturelle'' * ''Pani'' (Polish), translating as Mrs. * The ''Pacific Rep ...
,
p. 91 P. is an abbreviation or acronym that may refer to: * Page (paper), where the abbreviation comes from Latin ''pagina'' * Paris Herbarium, at the ''Muséum national d'histoire naturelle'' * ''Pani'' (Polish), translating as Mrs. * The ''Pacific Rep ...
,
p. 89 P. is an abbreviation or acronym that may refer to: * Page (paper), where the abbreviation comes from Latin ''pagina'' * Paris Herbarium, at the '' Muséum national d'histoire naturelle'' * ''Pani'' (Polish), translating as Mrs. * The ''Pacific Re ...
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*
Attribute domain Attribute may refer to: * Attribute (philosophy), an extrinsic property of an object * Attribute (research), a characteristic of an object * Grammatical modifier, in natural languages * Attribute (computing), a specification that defines a proper ...
*
Bijection, injection and surjection In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which '' arguments'' (input expressions from the domain) and '' images'' (output expressions from the codomain) are related or '' ...
*
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 ...
* Domain decomposition *
Effective domain In convex analysis, a branch of mathematics, the effective domain is an extension of the domain of a function defined for functions that take values in the extended real number line \infty, \infty= \mathbb \cup \. In convex analysis and variatio ...
* Image (mathematics) *
Lipschitz domain In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a 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 Lipschitz continuous function. The ...
* Naive set theory *
Support (mathematics) 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 smalle ...

# References

* {{Mathematical logic Functions and mappings Basic concepts in set theory