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| − | {{short description|mathematical concept}}
| + | ==Definition== |
| | + | In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. It is the set X in the notation f: X → Y, and is alternatively denoted as dom(f). Since a (total) function is defined on its entire domain, its domain coincides with its domain of definition. However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain. |
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| − | In [[mathematics]], the '''domain''' or '''set of departure''' of a [[Function (mathematics)|function]] is the [[Set (mathematics)|set]] into which all of the input of the function is constrained to fall.<ref name="Codd1970">{{cite journal |last1=Codd |first1=Edgar Frank |date=June 1970 |title=A Relational Model of Data for Large Shared Data Banks |url=https://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf |journal=Communications of the ACM |volume=13 |issue=6 |pages=377-387 |doi=10.1145/362384.362685 |access-date=2020-04-29}}</ref> It is the set {{mvar|X}} in the notation {{math|''f'': ''X'' → ''Y''}}, and is alternatively denoted as <math>\operatorname{dom}(f)</math>.<ref>{{Cite web|date=2020-03-01|title=Compendium of Mathematical Symbols|url=https://mathvault.ca/hub/higher-math/math-symbols/|access-date=2020-08-28|website=Math Vault|language=en-US}}</ref> Since a (total) function is defined on its entire domain, its domain coincides with its [[domain of definition]].<ref>{{cite book |last=Paley |first=Hiram |author-link=Hiram Paley |first2=Paul M. |last2=Weichsel |title=A First Course in Abstract Algebra |url=https://archive.org/details/firstcourseabstr00pale |url-access=limited |location=New York |publisher=Holt, Rinehart and Winston |year=1966 |page=[https://archive.org/details/firstcourseabstr00pale/page/n28 16] }}</ref> However, this coincidence is no longer true for a [[partial function]], since the domain of definition of a partial function can be a [[proper subset]] of the domain.
| + | A domain is part of a function f if f is defined as a triple (X, Y, G), where X is called the domain of f, Y its codomain, and G its graph. |
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| − | A domain is part of a function {{mvar|f}} if {{mvar|f}} is defined as a triple {{math|(''X'', ''Y'', ''G'')}}, where {{mvar|X}} is called the ''domain'' of {{mvar|f}}, {{mvar|Y}} its ''[[codomain]]'', and {{mvar|G}} its ''[[Graph of a function|graph]]''.<ref>{{Harvnb|Bourbaki|1970|p=76}}</ref> | + | A domain is not part of a function f if f is defined as just a graph. 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 → Y. |
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| − | A domain is not part of a function {{mvar|f}} if {{mvar|f}} is defined as just a graph.<ref>{{Harvnb|Bourbaki|1970|p=77}}</ref><ref>{{Harvnb|Forster|2003}}, [{{Google books|plainurl=y|id=mVeTuaRwWssC|page=10|text=Some mathematical cultures make this explicit, saying that a function}} pp. 10–11]</ref> For example, it is sometimes convenient in [[set theory]] to permit the domain of a function to be a [[Class (set theory)|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''}}.<ref>{{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|1967}}, [{{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|2004}}, [{{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]</ref>
| + | For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases). |
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| − | For instance, the domain of [[cosine]] is the set of all [[real numbers]], while the domain of the [[square root]] consists only of numbers greater than or equal to 0 (ignoring [[complex numbers]] in both cases).
| + | If the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the x-axis. |
| | + | The domain of a function f can be thought of as the set of all x values that can be plugged into f(x) that return a valid output. For example, if we have a function g(x) in the Cartesian plane, the domain is all of the x values such that g(x) is a real number. |
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| − | If the domain of a function is a subset of the real numbers and the function is represented in a [[Cartesian coordinate system]], then the domain is represented on the ''x''-axis.
| + | Examples: |
| | + | * Let <math> S </math> be a set of ordered pairs such that <math> S = \{(1,2), (2,3), (4, 7), (13, 9), (-20, 0)\}</math>. The domain is the set of all x values of <math> S </math>, so the domain is <math> \{-20, 1, 2, 4, 13\} </math>. |
| | + | * The domain of <math> g(x) = 1/x </math> is all real numbers EXCEPT 0, since 1/0 is not defined. |
| | + | * The domain of <math> h(x) = \sqrt{x} </math> is <math> [0, \inf) </math>, since <math> \sqrt{x} </math> is only defined when <math> x </math> is nonnegative (that is, when <math> x </math> is greater than or equal to 0). |
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| − | == Examples == | + | ==Resources== |
| − | A well-defined function must map every element of its domain to an element of its codomain. For example, the function <math>f</math> defined by
| + | * [https://www.youtube.com/watch?v=Q3NWljhiSJg Domain and Range: Basic Idea], patrickJMT |
| − | : <math>f(x)=\frac{1}{x}</math> | + | * [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-the-domain-of-a-function-defined-by-an-equation/ Finding Domain with an Equation], Lumen Learning |
| − | has no value for <math>f(0)</math>. Thus the set of all [[real number]]s, <math>\mathbb{R}</math>, cannot be its domain. In cases like this, the function is either defined on <math>\mathbb{R} \setminus \{ 0 \}</math>, or the "gap is plugged" by defining <math>f(0)</math> explicitly.
| + | * [https://courses.lumenlearning.com/ivytech-collegealgebra/chapter/find-domain-and-range-from-graphs/ Finding Domain and Range with Graphs], Lumen Learning |
| − | For example. if one extends the definition of <math>f</math> to the [[piecewise]] function
| + | * [https://www.youtube.com/watch?v=w81y25anEOM Finding the Domain Algebraically], patrickJMT |
| − | : <math>f(x) = \begin{cases} | + | * [https://www.youtube.com/watch?v=BxaYyS6lsQ4 Finding Domain and Range of a Piecewise Function], patrickJMT |
| − | 1/x&x\not=0\\
| + | * [https://mathculus.com/how-to-find-the-domain-of-a-function-algebraically/ How to Find Domain + Example Problems], Math Culus |
| − | 0&x=0
| |
| − | \end{cases}</math>
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| − | then ''<math>f</math>'' is defined for all real numbers, and its domain is <math>\mathbb{R}</math>.
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| − | Any function can be restricted to a subset of its domain. The [[Restriction (mathematics)|restriction]] of <math>g \colon A \to B</math> to <math>S</math>, where <math>S \subseteq A</math>, is written as <math>\left. g \right|_{S} \colon S \to B</math>.
| + | == Licensing == |
| − | | + | Content obtained and/or adapted from: |
| − | == Natural domain == | + | * [https://en.wikipedia.org/wiki/Domain_of_a_function Domain of a Function, Wikipedia] under a CC BY-SA license |
| − | The '''natural domain''' of a function (sometimes shortened as domain) is the maximum set of values for which the function is defined, typically within the reals but sometimes among the integers or complex numbers as well. For instance, the natural domain of square root is the non-negative reals when considered as a real number function. When considering a natural domain, the set of possible values of the function is typically called its [[Range of a function|range]].<ref>{{cite book|title=Calculus: basic concepts and applications|url=https://archive.org/details/calculusbasiccon00rose_328|url-access=limited|first1=Robert A.|last1=Rosenbaum|first2=G. Philip|last2=Johnson|page=[https://archive.org/details/calculusbasiccon00rose_328/page/n76 60]|year=1984|isbn=0-521-25012-9|publisher=Cambridge University Press}}</ref> Also, in [[complex analysis]] especially [[Function of several complex variables|several complex variables]], when a function ''f'' is [[Holomorphic function|holomorpic]] on the domain <math>D\subset \Complex^n</math> and cannot directly connect to the domain outside ''D'', including the point of the domain boundary <math>\partial D</math>, in other words, such a domain ''D'' is a '''natural domain''' in the sense of [[analytic continuation]], the domain ''D'' is called the [[domain of holomorphy]] of ''f'' and the boundary is called the natural boundary of ''f''.
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| − | == Category theory ==
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| − | [[Category theory]] deals with [[morphisms]] instead of functions. Morphisms are arrows from one object to another. The domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned—or at least formulated more abstractly. For example, the notion of restricting a morphism to a subset of its domain must be modified. For more, see [[subobject]].
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| − | == Other uses ==
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| − | {{main|Domain (mathematical analysis)}}
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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 space|connected]] [[open set]].<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Domain|url=https://mathworld.wolfram.com/Domain.html|access-date=2020-08-28|website=mathworld.wolfram.com|language=en}}</ref> In [[real analysis|real]] and [[complex analysis]], a domain is an [[open set|open]] [[connected space|connected]] subset of a [[real number|real]] or [[complex number|complex]] vector space. In the study of [[partial differential equation]]s, 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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| − | == More common examples ==
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| − | As a partial function from the real numbers to the real numbers, the function <math> x\mapsto \sqrt{x}</math> has domain <math>x \geq 0</math>. However, if one defines the square root of a negative number ''x'' as the [[complex number]] ''z'' with positive [[imaginary part]] such that ''z''<sup>2</sup> = ''x'', then the function <math> x\mapsto \sqrt{x}</math> has the entire real line as its domain (but now with a larger codomain).
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| − | The domain of the [[trigonometric function]] <math>\tan x = \tfrac{\sin x}{\cos x}</math> is the set of all (real or complex) numbers, that are not of the form <math>\tfrac{\pi}{2} + k \pi, k = 0, \pm 1, \pm 2, \ldots</math>.
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| − | == See also ==
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| − | * [[Attribute domain]]
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| − | * [[Bijection, injection and surjection]]
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| − | * [[Codomain]]
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| − | * [[Domain decomposition]]
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| − | * [[Effective domain]]
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| − | * [[Image (mathematics)]]
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| − | * [[Lipschitz domain]]
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| − | * [[Naive set theory]]
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| − | * [[Support (mathematics)]]
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| − | == Notes ==
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| − | {{Reflist}}
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| − | == References ==
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| − | * {{cite book |last=Bourbaki |first=Nicolas |title=Théorie des ensembles |year=1970 |publisher=Springer |series=Éléments de mathématique |isbn=9783540340348}}
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| − | {{Mathematical logic}}
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| − | [[Category:Functions and mappings]]
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| − | [[Category:Basic concepts in set theory]]
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Definition
In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall. It is the set X in the notation f: X → Y, and is alternatively denoted as dom(f). Since a (total) function is defined on its entire domain, its domain coincides with its domain of definition. However, this coincidence is no longer true for a partial function, since the domain of definition of a partial function can be a proper subset of the domain.
A domain is part of a function f if f is defined as a triple (X, Y, G), where X is called the domain of f, Y its codomain, and G its graph.
A domain is not part of a function f if f is defined as just a graph. 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 → Y.
For instance, the domain of cosine is the set of all real numbers, while the domain of the square root consists only of numbers greater than or equal to 0 (ignoring complex numbers in both cases).
If the domain of a function is a subset of the real numbers and the function is represented in a Cartesian coordinate system, then the domain is represented on the x-axis.
The domain of a function f can be thought of as the set of all x values that can be plugged into f(x) that return a valid output. For example, if we have a function g(x) in the Cartesian plane, the domain is all of the x values such that g(x) is a real number.
Examples:
- Let
be a set of ordered pairs such that 
. The domain is the set of all x values of , so the domain is 
.
- The domain of
is all real numbers EXCEPT 0, since 1/0 is not defined.
- The domain of
is 
, since 
is only defined when 
is nonnegative (that is, when is greater than or equal to 0).
Resources
Licensing
Content obtained and/or adapted from:
