Domain of a function

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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 ...
, the domain of 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 the
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 ...
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. 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 th ...
, 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 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 ...
. Any function can be restricted to a subset of its domain. 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 $f \colon X \to Y$ 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 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 ...
, 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 In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
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 language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a domain is a
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
. In
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ... and complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ... , a domain is an open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ... connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ... subset of a real Real may refer to: Currencies * Brazilian real (R$) * 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 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 ...
vector space. In the study of
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, 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, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
$\mathbb^$ 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 conce ...
to permit the domain of a function to be a proper class , 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 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 ...
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)". C ...
; ,
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 Repo ...
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 Repo ...
,
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 Repo ...
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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 prope ...
*
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 ''m ...
*
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 th ...
*
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 soluti ...
*
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) 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) ...
*
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. T ...
*
Naive set theory Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike Set theory#Axiomatic set theory, axiomatic set theories, which are defined using Mathematical_logic#Formal_logical_systems, forma ...
*
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