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Domain of a function

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From Wikipedia, the free encyclopedia

Set of all things that may be the input of a mathematical function

A functionf fromX toY. The set of points in the red ovalX is the domain off.The set of points in the blue ovalY is the codomain off.The set of points in the yelllow oval is the range off.

Graph of the arcsine and arccosine functions,f(x) = arcsin(x) andf(x) = arccos(x), each of whose domain consists of the set of real numbers [–1,1] inclusively

Inmathematics, thedomain of a function is theset of inputs accepted by thefunction. It is sometimes denoted by $\operatorname {dom} (f)$ or $\operatorname {dom} f$ , wheref is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".^[1]

More precisely, given a function $f\colon X\to Y$ , the domain off isX. In modern mathematical language, the domain is part of the definition of a function rather than a property of it.

In the special case thatX andY are both sets ofreal numbers, the functionf can be graphed in theCartesian coordinate system. In this case, the domain is represented on thex-axis of the graph, as the projection of the graph of the function onto thex-axis.

For a function $f\colon X\to Y$ , the setY is called thecodomain: the set to which all outputs must belong. The set of specific outputs the function assigns to elements ofX is called itsrange orimage. The image of $f {\displaystyle f}$ is a subset ofY, shown as the yellow oval in the accompanying diagram.

Any function can be restricted to a subset of its domain. Therestriction of $f\colon X\to Y$ to $A {\displaystyle A}$ , where $A\subseteq X$ , is written as $\left.f\right|_{A}\colon A\to Y$ .

Natural domain

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If areal functionf is given by a formula, it may be not defined for some values of the variable. In this case, it is apartial function, and the set of real numbers on which the formula can be evaluated to a real number is called thenatural domain ordomain of definition off. In many contexts, a partial function is called simply afunction, and its natural domain is called simply itsdomain.

Examples

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The function $f {\displaystyle f}$ defined by $f(x)={\frac {1}{x}}$ cannot be evaluated at 0. Therefore, the natural domain of $f {\displaystyle f}$ is the set of real numbers excluding 0, which can be denoted by $\mathbb {R} \setminus \{0\}$ or $\{x\in \mathbb {R} :x\neq 0\}$ .
Thepiecewise function $f {\displaystyle f}$ 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.
Thesquare 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 $\{x\in \mathbb {R} :x\geq 0\}$ .
Thetangent 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 someinteger $k {\displaystyle k}$ , which can be written as $\mathbb {R} \setminus \{{\tfrac {\pi }{2}}+k\pi :k\in \mathbb {Z} \}$ .

Other uses

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The termdomain is also commonly used in a different sense inmathematical analysis: adomain is anon-empty connected open set in atopological space. In particular, inreal andcomplex analysis, adomain is a non-empty connected open subset of thereal coordinate space $\mathbb {R} ^{n}$ or thecomplex coordinate space $\mathbb {C} ^{n}.$

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 ofpartial differential equations: in that case, adomain is the open connected subset of $\mathbb {R} ^{n}$ 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

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For example, it is sometimes convenient inset theory to permit the domain of a function to be aproper classX, in which case there is formally no such thing as a triple(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 formf:X →Y.^[2]

Notes

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^"Domain, Range, Inverse of Functions".Easy Sevens Education. 10 April 2023. Retrieved2023-04-13.
^Eccles 1997, p. 91 (quote 1,quote 2);Mac Lane 1998,p. 8; Mac Lane, inScott & Jech 1971,p. 232;Sharma 2010,p. 91;Stewart & Tall 1977,p. 89

References

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

Mathematical logic

General

Theorems
(list),
paradoxes

Logics

Traditional	Classical logic Logical truth Tautology Proposition Inference Logical equivalence Consistency Equiconsistency Argument Soundness Validity Syllogism Square of opposition Venn diagram
Propositional	Boolean algebra Boolean functions Logical connectives Propositional calculus Propositional formula Truth tables Many-valued logic 3 finite ∞
Predicate	First-order list Second-order Monadic Higher-order Fixed-point Free Quantifiers Predicate Monadic predicate calculus

Set theory

Set hereditary Class (Ur-)Element Ordinal number Extensionality Forcing Relation equivalence partition Set operations: intersection union complement Cartesian product power set identities
Types ofsets	Countable Uncountable Empty Inhabited Singleton Finite Infinite Transitive Ultrafilter Recursive Fuzzy Universal Universe constructible Grothendieck Von Neumann
Maps, cardinality	Function/Map domain codomain image In/Sur/Bi-jection Schröder–Bernstein theorem Isomorphism Gödel numbering Enumeration Large cardinal inaccessible Aleph number Operation binary
Theories	Zermelo–Fraenkel axiom of choice continuum hypothesis General Kripke–Platek Morse–Kelley Naive New Foundations Tarski–Grothendieck Von Neumann–Bernays–Gödel Ackermann Constructive