Difference between revisions of "Domain"

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{{short description|Mathematical concept}}
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[[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}}.]]
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[[File:Square_root_0_25.svg|thumb|250px|Graph of the real-valued square root function, ''f''(''x'') = {{radic|''x''}}, whose domain consists of all nonnegative real numbers]]
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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>, where {{math|''f''}} is the function.
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More precisely, given a function <math>f\colon X\to Y</math>, the domain of {{math|''f''}} is {{math|''X''}}. Note that in modern mathematical language, the domain is part of the definition of a function rather than a property of it.
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In the special case that {{math|''X''}} and {{math|''Y''}} are both subsets of <math>\R</math>, 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.
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For a function <math>f\colon X\to Y</math>, the set {{math|''Y''}} is called the codomain, and the set of values attained by the function (which is a subset of {{math|''Y''}}) is called its range or image.
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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>.
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== Natural domain ==
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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''.
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=== Examples ===
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* The function <math>f</math> defined by <math>f(x)=\frac{1}{x}</math> cannot be evaluated at 0. Therefore the natural domain of {{mvar|f}} is <math>\mathbb{R} \setminus \{ 0 \}</math>.
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* In contrast, if <math>f</math> is the piecewise function <math>f(x) = \begin{cases}
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1/x&x\not=0\\
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0&x=0
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\end{cases},</math> then ''<math>f</math>'' is defined for all real numbers, and its natural domain is <math>\mathbb{R}</math>.
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* The function  <math>x\mapsto\sqrt x</math> has as its natural domain the non-negative real numbers, which can be denoted by <math>\mathbb R_{\geq 0}</math>, by the interval <math>(0,\infty)</math>, or by <math>\{x\in\mathbb R:x\geq 0\}</math>.
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* The tangent function <math>\tan x</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> where {{mvar|k}} is any integer.
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== Other uses ==
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The word "domain" is used with other related meanings in some areas of mathematics. In topology, a domain is a connected open set. In real and complex analysis, a domain is an open connected subset of a real or complex vector space. In the study of partial differential equations, a domain is the open connected subset of the Euclidean space <math>\mathbb{R}^{n}</math> where a problem is posed (i.e., where the unknown function(s) are defined).
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== Set theoretical notions ==
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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''}}.
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== Resources ==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_Functions.pdf Domain Range and Toolkit Functions], Book Chapter
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_Functions.pdf Domain Range and Toolkit Functions], Book Chapter
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_FunctionsGN.pdf Guided Notes]
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1053/Domain_Range_and_Toolkit_Functions/MAT1053_M1.2Domain_Range_and_Toolkit_FunctionsGN.pdf Guided Notes]
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== Licensing ==
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Content obtained and/or adapted from:
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* [https://en.wikipedia.org/wiki/Domain_of_a_function Domain of a function, Wikipedia] under a CC BY-SA license

Revision as of 17:55, 9 January 2022

[[Category:Template:Pagetype with short description]]Expression error: Unexpected < operator.

A function f from X to Y. The set of points in the red oval X is the domain of f.
Graph of the real-valued square root function, f(x) = Template:Radic, 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 , where f is the function.

More precisely, given a function , the domain of f is X. Note that 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 subsets of , 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 , the set Y is called the codomain, and the set of values attained by the function (which is a subset of Y) is called its range or image.

Any function can be restricted to a subset of its domain. The restriction of to , where , is written as .

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 defined by cannot be evaluated at 0. Therefore the natural domain of f is .
  • In contrast, if is the piecewise function then is defined for all real numbers, and its natural domain is .
  • The function has as its natural domain the non-negative real numbers, which can be denoted by , by the interval , or by .
  • The tangent function has as its natural domain the set of all real numbers which are not of the form where k is any integer.

Other uses

The word "domain" is used with other related meanings in some areas of mathematics. In topology, a domain is a connected open set. In real and complex analysis, a domain is an open connected subset of a real or complex vector space. In the study of partial differential equations, a domain is the open connected subset of the Euclidean space where a problem is posed (i.e., where the unknown function(s) are defined).

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: XY.

Resources

Licensing

Content obtained and/or adapted from: