GetWiki

{{short description|Mathematical concept}}(File:Codomain2.SVG|right|thumb|250px|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}}.)File:Square_root_0_25.svg|thumb|250px|Graph of the real-valued x}}, whose domain consists of all nonnegative real numbersIn mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by operatorname{dom}(f) or operatorname{dom }f, 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”.WEB, Domain, Range, Inverse of Functions,www.easysevens.com/domain-range-inverse-of-functions/, 2023-04-13, Easy Sevens Education, en, More precisely, given a function fcolon Xto Y, 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 fcolon Xto Y, 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 f 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 f colon X to Y to A, where Asubseteq X, is written as left. f right|_A colon A to Y.

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 f defined by f(x)=frac{1}{x} 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{R} setminus { 0 } or {xinmathbb R:xne 0}.
• The piecewise function f defined by f(x) = begin{cases}

1/x&xnot=0end{cases}, has as its natural domain the set mathbb{R} of real numbers.

• The square root function f(x)=sqrt x has as its natural domain the set of non-negative real numbers, which can be denoted by mathbb R_{geq 0}, the interval [0,infty), or {xinmathbb R:xgeq 0}.
• The tangent function, denoted tan, has as its natural domain the set of all real numbers which are not of the form tfrac{pi}{2} + k pi for some integer k, which can be written as mathbb R setminus {tfrac{pi}{2}+kpi: kinmathbb Z}.

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 R^n or the complex coordinate space C^n.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 R^{n} 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}}.{{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]

See also

• Argument of a function
• Attribute domain
• Bijection, injection and surjection
• Codomain
• Domain decomposition
• Effective domain
• Image (mathematics)
• Lipschitz domain
• Naive set theory
• Range of a function
• Support (mathematics)

Notes

{{Reflist}}

References

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

{{Mathematical logic}}