Domain of a function

A function

f

from

X

to

Y

. The set of points in the red oval

X

is the domain of

f

. The set of points in the blue oval

Y

is the codomain of

f

. The set of points in the yellow oval is the range of

f

.

File:Arcsine Arccosine.svg

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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \operatorname {dom} (f)} or Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \operatorname {dom} f} , where $f$ 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\colon X\to Y} , the domain of $f$ is $X$ . 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 $X$ and $Y$ are both sets of real numbers, the function $f$ can be graphed in the Cartesian coordinate system. In this case, the domain is represented on the $x$ -axis of the graph, as the projection of the graph of the function onto the $x$ -axis.

For a function Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\colon X\to Y} , the set $Y$ is called the codomain: the set to which all outputs must belong. The set of specific outputs the function assigns to elements of $X$ is called its range or image. The image of $f$ is a subset of $Y$ , shown as the yellow oval in the accompanying diagram.

Any function can be restricted to a subset of its domain. The restriction of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\colon X\to Y} to $A$ , where $A\subseteq X$ , is written as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.f\right|_{A}\colon A\to Y} .

Natural domain

If a real function $f$ 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 $f$ . 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\frac{1}{x}} cannot be evaluated at 0. Therefore, the natural domain of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is the set of real numbers excluding 0, which can be denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R} \setminus \{ 0 \}} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x\in\mathbb R:x\ne 0\}} .
The piecewise function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \begin{cases} 1/x&x\not=0\\ 0&x=0 \end{cases},} has as its natural domain the set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{R}} of real numbers.
The square root function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\sqrt x} has as its natural domain the set of non-negative real numbers, which can be denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb R_{\geq 0}} , the interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,\infty)} , or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{x\in\mathbb R:x\geq 0\}} .
The tangent function, denoted Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan} , has as its natural domain the set of all real numbers which are not of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tfrac{\pi}{2} + k \pi} for some integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} , which can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R^n} or the complex coordinate space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 $X$ , 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 form $f : X \to Y$ .^[2]

Notes

↑ "Domain, Range, Inverse of Functions". Easy Sevens Education. 10 April 2023. Retrieved 2023-04-13.
↑ Eccles 1997, p. 91 ([[[:Template:Google books]] quote 1], [[[:Template:Google books]] quote 2]); Mac Lane 1998, [[[:Template:Google books]] p. 8]; Mac Lane, in Scott & Jech 1971, [[[:Template:Google books]] p. 232]; Sharma 2010, [[[:Template:Google books]] p. 91]; Stewart & Tall 1977, [[[:Template:Google books]] p. 89]

References

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.

Template:Mathematical logic

[1] "Domain, Range, Inverse of Functions". Easy Sevens Education. 10 April 2023. Retrieved 2023-04-13.

[2] Eccles 1997, p. 91 ([[[:Template:Google books]] quote 1], [[[:Template:Google books]] quote 2]); Mac Lane 1998, [[[:Template:Google books]] p. 8]; Mac Lane, in Scott & Jech 1971, [[[:Template:Google books]] p. 232]; Sharma 2010, [[[:Template:Google books]] p. 91]; Stewart & Tall 1977, [[[:Template:Google books]] p. 89]

[1]

[2]

Domain of a function

Page version status

Contents

Natural domain

Examples

Other uses

Set theoretical notions

See also

Notes

References

Navigation menu

Domain of a function

Natural domain

Examples

Other uses

Set theoretical notions

See also

Notes

References

Navigation menu

Search