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{{Short description|Subset of a function's codomain}}
The '''range of a function''' may refer to either:
{{For|the statistical concept|Range (statistics)}}[[Image:Codomain2.SVG|right|thumb|350px|<math>f</math> is a function from [[domain of a function|domain]] '''''X''''' to [[codomain]] '''''Y'''''. The yellow oval inside '''''Y''''' is the [[Image (mathematics)|image]] of <math>f</math>. Sometimes "range" refers to the image and sometimes to the codomain.]]

In [[mathematics]], the '''range of a function''' may refer to either of two closely related concepts:

* The [[codomain]] of the [[Function (mathematics)|function]]
* The [[codomain]] of the [[Function (mathematics)|function]]
* The [[Image (mathematics)|image]] of the function
* The [[Image (mathematics)|image]] of the function


{{dab}}
Given two [[set (mathematics)|set]]s {{mvar|X}} and {{mvar|Y}}, a [[binary relation]] {{mvar|f}} between {{mvar|X}} and {{mvar|Y}} is a (total) function (from {{mvar|X}} to {{mvar|Y}}) if for every {{mvar|x}} in {{mvar|X}} there is exactly one {{mvar|y}} in {{mvar|Y}} such that {{mvar|f}} relates {{mvar|x}} to {{mvar|y}}. The sets {{mvar|X}} and {{mvar|Y}} are called [[domain of a function|domain]] and codomain of {{mvar|f}}, respectively. The image of {{mvar|f}} is then the [[subset]] of {{mvar|Y}} consisting of only those [[Element (mathematics)|element]]s {{mvar|y}} of {{mvar|Y}} such that there is at least one {{mvar|x}} in {{mvar|X}} with {{math|1=''f''(''x'') = ''y''}}.

==Terminology==
As the term "range" can have different meanings, it is considered a good practice to define it the first time it is used in a textbook or article. Older books, when they use the word "range", tend to use it to mean what is now called the [[codomain]].{{sfnm|1a1=Hungerford|1y=1974|1p=3|2a1=Childs|2y=2009|2p=140}} More modern books, if they use the word "range" at all, generally use it to mean what is now called the [[image (mathematics)|image]].{{sfn|Dummit|Foote|2004|p=2}} To avoid any confusion, a number of modern books don't use the word "range" at all.{{sfn|Rudin|1991|p=99}}

==Elaboration and example==
Given a function
:<math>f \colon X \to Y</math>
with [[domain of a function|domain]] <math>X</math>, the range of <math>f</math>, sometimes denoted <math>\operatorname{ran}(f)</math> or <math>\operatorname{Range}(f)</math>,<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Range|url=https://mathworld.wolfram.com/Range.html|access-date=2020-08-28|website=mathworld.wolfram.com|language=en}}</ref> may refer to the codomain or target set <math>Y</math> (i.e., the set into which all of the output of <math>f</math> is constrained to fall), or to <math>f(X)</math>, the image of the domain of <math>f</math> under <math>f</math> (i.e., the subset of <math>Y</math> consisting of all actual outputs of <math>f</math>). The image of a function is always a subset of the codomain of the function.<ref>{{Cite web|last=Nykamp|first=Duane|date=|title=Range definition|url=https://mathinsight.org/definition/range|archive-url=|archive-date=|access-date=August 28, 2020|website=Math Insight}}</ref>

As an example of the two different usages, consider the function <math>f(x) = x^2</math> as it is used in [[real analysis]] (that is, as a function that inputs a [[real number]] and outputs its square). In this case, its codomain is the set of real numbers <math>\mathbb{R}</math>, but its image is the set of non-negative real numbers <math>\mathbb{R}^+</math>, since <math>x^2</math> is never negative if <math>x</math> is real. For this function, if we use "range" to mean ''codomain'', it refers to <math>\mathbb{{\displaystyle \mathbb {R} ^{}}}</math>; if we use "range" to mean ''image'', it refers to <math>\mathbb{R}^+</math>.

In many cases, the image and the codomain can coincide. For example, consider the function <math>f(x) = 2x</math>, which inputs a real number and outputs its double. For this function, the codomain and the image are the same (both being the set of real numbers), so the word range is unambiguous.

==See also==
* [[Bijection, injection and surjection]]
* [[Essential range]]

==Notes and references==

{{Reflist}}

==Bibliography==

*{{Cite book
| first = Lindsay N.
| last = Childs
| editor-first1 = Lindsay N.
| editor-last1 = Childs
| title = A Concrete Introduction to Higher Algebra
| series = [[Undergraduate Texts in Mathematics]]
| edition = 3rd
| publisher = Springer
| year = 2009
| isbn = 978-0-387-74527-5
| oclc = 173498962
| doi = 10.1007/978-0-387-74725-5
}}
*{{Cite book
| first1 = David S.
| last1 = Dummit
| first2 = Richard M.
| last2 = Foote
| title = Abstract Algebra
| edition = 3rd
| publisher = Wiley
| year = 2004
| isbn = 978-0-471-43334-7
| oclc = 52559229
}}
*{{Cite book
| first = Thomas W.
| last = Hungerford
| author-link = Thomas W. Hungerford
| title = Algebra
| publisher = Springer
| series = [[Graduate Texts in Mathematics]]
| volume = 73
| year = 1974
| isbn = 0-387-90518-9
| oclc = 703268
| doi = 10.1007/978-1-4612-6101-8
}}
*{{Cite book
| first = Walter
| last = Rudin
| title = Functional Analysis
| edition = 2nd
| publisher = McGraw Hill
| year = 1991
| isbn = 0-07-054236-8
| url-access = registration
| url = https://archive.org/details/functionalanalys00rudi
}}

{{Mathematical logic}}

{{DEFAULTSORT:Range (Mathematics)}}
[[Category:Functions and mappings]]
[[Category:Basic concepts in set theory]]

Revision as of 23:58, 1 December 2023

The range of a function may refer to either:

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