Domain of a function

File:Square_root_0_25.svg function, f(x) = {{radic|x}}, whose domain consists of all nonnegative real numbers]]

In 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".{{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}}

More precisely, given a function $f\colon X\to Y$ , 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 $f\colon X\to Y$ , 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 $A\subseteq X$ , 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 $\{x\in\mathbb R:x\ne 0\}$ .
The piecewise function $f$ defined by $f(x) = \begin{cases}$

1/x&x\not=0\\ 0&x=0 \end{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 $\{x\in\mathbb R:x\geq 0\}$ .
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 $\mathbb R \setminus \{\tfrac{\pi}{2}+k\pi: k\in\mathbb Z\}$ .

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]

Notes

References

{{cite book |last=Bourbaki |first=Nicolas |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 |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