Domain of a function

{{Short description|Set of all things that may be the input of a mathematical function}} thumb|A function {{mvar|f}} from {{mvar|X}} to {{mvar|Y}}. The set of points in the red oval {{mvar|X}} is the domain of {{mvar|f}}. The set of points in the blue oval {{mvar|Y}} is the codomain of {{mvar|f}}. The set of points in the yellow oval is the range of {{mvar|f}}. [[File:Arcsine Arccosine.svg|thumb|upright=0.75|Graph of the arcsine and arccosine functions, ''f''(''x'') = arcsin(''x'') and ''f''(''x'') = arccos(''x''), each of whose domain consists of the set of real numbers [–1,1] inclusively]]

In mathematics, the '''domain of a function''' is the set of inputs accepted by the function. It is sometimes denoted by <math>\operatorname{dom}(f)</math> or <math>\operatorname{dom }f</math>, where {{math|''f''}} is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".<ref>{{Cite web|title=Domain, Range, Inverse of Functions|url=https://www.easysevens.com/domain-range-inverse-of-functions/|access-date=2023-04-13|website=Easy Sevens Education|date=10 April 2023 |language=en}}</ref>

More precisely, given a function <math>f\colon X\to Y</math>, the domain of {{math|''f''}} is {{math|''X''}}. 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 {{math|''X''}} and {{math|''Y''}} are both sets of real numbers, the function {{math|''f''}} can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the {{math|''x''}}-axis of the graph, as the projection of the graph of the function onto the {{math|''x''}}-axis.

For a function <math>f\colon X\to Y</math>, the set {{math|''Y''}} is called the ''codomain'': the set to which all outputs must belong. The set of specific outputs the function assigns to elements of {{math|''X''}} is called its ''range'' or ''image''. The image of <math>f</math> is a subset of {{math|''Y''}}, shown as the yellow oval in the accompanying diagram.

Any function can be restricted to a subset of its domain. The restriction of <math>f \colon X \to Y</math> to <math>A</math>, where <math>A\subseteq X</math>, is written as <math>\left. f \right|_A \colon A \to Y</math>.

== Natural domain == If a real function {{mvar|f}} 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 {{mvar|f}}. In many contexts, a partial function is called simply a ''function'', and its natural domain is called simply its ''domain''.

=== Examples ===

* The function <math>f</math> defined by <math>f(x)=\frac{1}{x}</math> cannot be evaluated at 0. Therefore, the natural domain of <math>f</math> is the set of real numbers excluding 0, which can be denoted by <math>\mathbb{R} \setminus \{ 0 \}</math> or <math>\{x\in\mathbb R:x\ne 0\}</math>. * The piecewise function <math>f</math> defined by <math>f(x) = \begin{cases} 1/x&x\not=0\\ 0&x=0 \end{cases},</math> has as its natural domain the set <math>\mathbb{R}</math> of real numbers. * The square root function <math>f(x)=\sqrt x</math> has as its natural domain the set of non-negative real numbers, which can be denoted by <math>\mathbb R_{\geq 0}</math>, the interval <math>[0,\infty)</math>, or <math>\{x\in\mathbb R:x\geq 0\}</math>. * The tangent function, denoted <math>\tan</math>, has as its natural domain the set of all real numbers which are not of the form <math>\tfrac{\pi}{2} + k \pi</math> for some integer <math>k</math>, which can be written as <math>\mathbb R \setminus \{\tfrac{\pi}{2}+k\pi: k\in\mathbb Z\}</math>.

== Other uses ==

The term ''domain'' is also commonly used in a different sense in mathematical analysis: a ''domain'' is a non-empty connected open set in a topological space. In particular, in real and complex analysis, a ''domain'' is a non-empty connected open subset of the real coordinate space <math>\R^n</math> or the complex coordinate space <math>\C^n.</math>

Sometimes such a domain is used as the domain of a function, although functions may be defined on more general sets. The two concepts are sometimes conflated as in, for example, the study of partial differential equations: in that case, a ''domain'' is the open connected subset of <math>\R^{n}</math> where a problem is posed, making it both an analysis-style domain and also the domain of the unknown function(s) sought.

== Set theoretical notions == For example, it is sometimes convenient in set theory to permit the domain of a function to be a proper class {{mvar|X}}, in which case there is formally no such thing as a triple {{math|(''X'', ''Y'', ''G'')}}. With such a definition, functions do not have a domain, although some authors still use it informally after introducing a function in the form {{math|''f'': ''X'' → ''Y''}}.<ref>{{Harvnb|Eccles|1997}}, p. 91 ([{{Google books|plainurl=y|id=ImCSX_gm40oC|page=91|text=The reader may wonder at this variety of ways of thinking about a function}} quote 1], [{{Google books|plainurl=y|id=ImCSX_gm40oC|page=91|text=When defining a function using a formula it is important to be clear about which sets are the domain and the codomain of the function}} quote 2]); {{Harvnb|Mac Lane|1998}}, [{{Google books|plainurl=y|id=MXboNPdTv7QC|page=8|text=Here "function" means a function with specified domain and specified codomain}} p. 8]; Mac Lane, in {{Harvnb|Scott|Jech|1971}}, [{{Google books|plainurl=y|id=5mf4Vckj0gEC|page=232|text=Note explicitly that the notion of function is not that customary in axiomatic set theory}} p. 232]; {{Harvnb|Sharma|2010}}, [{{Google books|plainurl=y|id=IGvDpe6hYiQC|page=91|text=Functions as sets of ordered pairs}} p. 91]; {{Harvnb|Stewart|Tall|1977}}, [{{Google books|plainurl=y|id=TLelvnIU2sEC|page=89|text=Strictly speaking we cannot talk of 'the' codomain of a function}} p. 89]</ref>

== Notes == {{Reflist}}

== References == * {{cite book |last=Bourbaki |first=Nicolas |author-link=Nicolas Bourbaki |title=Théorie des ensembles |year=1970 |publisher=Springer |series=Éléments de mathématique |isbn=9783540340348}} * {{cite book |last1=Eccles |first1=Peter J. |title=An Introduction to Mathematical Reasoning: Numbers, Sets and Functions |date=11 December 1997 |publisher=Cambridge University Press |isbn=978-0-521-59718-0 |url=https://books.google.com/books?id=ImCSX_gm40oC |language=en}} * {{cite book |last1=Mac Lane |first1=Saunders |author-link=Saunders Mac Lane |title=Categories for the Working Mathematician |date=25 September 1998 |publisher=Springer Science & Business Media |isbn=978-0-387-98403-2 |url=https://books.google.com/books?id=MXboNPdTv7QC |language=en}} * {{cite book |last1=Scott |first1=Dana S. |last2=Jech |first2=Thomas J. |title=Axiomatic Set Theory, Part 1 |date=31 December 1971 |publisher=American Mathematical Soc. |isbn=978-0-8218-0245-8 |url=https://books.google.com/books?id=5mf4Vckj0gEC |language=en}} * {{cite book |last1=Sharma |first1=A. K. |title=Introduction To Set Theory |date=2010 |publisher=Discovery Publishing House |isbn=978-81-7141-877-0 |url=https://books.google.com/books?id=IGvDpe6hYiQC |language=en}} * {{cite book |last1=Stewart |first1=Ian |last2=Tall |first2=David |title=The Foundations of Mathematics |date=1977 |publisher=Oxford University Press |isbn=978-0-19-853165-4 |url=https://books.google.com/books?id=TLelvnIU2sEC |language=en}}

Category:Functions and mappings Category:Basic concepts in set theory