Difference between revisions of "Range (mathematics)"

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{{Other uses|Range (disambiguation){{!}}Range}}
{{Other uses|Range (disambiguation){{!}}Range}}


[[Image:Codomain2.SVG|right|thumb|250px|<math>f</math> is a function from [[domain of a function|domain]] '''X''' to [[codomain]] '''Y'''. The smaller oval inside '''Y''' is the [[image (mathematics)|image]] of <math>f</math>.  Sometimes "range" refers to the codomain and sometimes to the image.]]
[[Image:Codomain2.SVG|right|thumb|250px|<math>f</math> is a function from [[domain of a function|domain]] '''X''' to [[codomain]] '''Y'''. The smaller 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 (mathematics)|function]] refers to either the '''[[codomain]]''' or the '''[[image (mathematics)|image]]''' of the function, depending upon usage. The codomain is a set containing the function's outputs, whereas the image is the part of the codomain which consists only of the function's outputs.  
In [[mathematics]], and more specifically in [[naive set theory]], the '''range''' of a [[Function (mathematics)|function]] refers to either the ''[[codomain]]'' or the ''[[image (mathematics)|image]]'' of the function, depending upon usage. Modern usage almost always uses ''range'' to mean ''image''.  


For example, the function <math>f(x) = x^2</math> is often described as a function from the [[real number]]s to the real numbers, meaning that its codomain is the set of real numbers '''R''', but its image is the set of non-negative real numbers, as <math>x^2</math> is never negative if <math>x</math> is real. Some books use the term range to indicate the codomain '''R'''. These books call the actual output of the function the image. This is the current usage for [[Range (computer science)|range in computer science]]. Other books use the term range to indicate the image, that is the non-negative real numbers. In this case, the larger set containing the range is called the codomain.<ref>Walter Rudin, ''Functional Analysis'', Second edition, p. 99, McGraw Hill, 1991, ISBN 0-07-054236-8</ref> This usage is more common in modern mathematics.  
The codomain of a function is some arbitrary set.  In [[real analysis]], it is the [[real number]]s. In [[complex analysis]], it is the [[complex number]]s.


==Examples==
The image of a function is the set of all outputs of the functionThe image is always a subset of the codomain.
Let ''f'' be a function on the [[real numbers]] <math>f\colon \mathbb{R}\rightarrow\mathbb{R}</math> defined by <math>f(x) = 2x</math>. This function takes as input any real number and outputs a real number two times the inputIn this case, the codomain and the image are the same (i.e. the function is a [[Surjective function|surjection]]), so the range is unambiguous; it is the set of all real numbers.


In contrast, consider the function <math>f\colon \mathbb{R}\rightarrow\mathbb{R}</math> defined by <math>f(x) = \sin(x)</math>. If the word "range" is used in the first sense given above, we would say the range of ''f'' is the codomain, all real numbers; but since the output of the [[sine]] function is always between −1 and 1, "range" in the second sense would say the range is the image, the closed interval from −1 to 1.
==Distinguishing between the two uses==
 
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]].<ref>Hungerford 1974, page 3.</ref><ref>Childs 1990, page 140.</ref>  More modern books, if they use the word "range" at all, generally use it to mean what is now called the [[image (mathematics)|image]].<ref>Dummit and Foote 2004, page 2.</ref>  To avoid any confusion, a number of modern books don't use the word "range" at all.<ref>Rudin 1991, page 99.</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{R}</math>.  When we use "range" to mean ''image'', it refers to <math>\mathbb{R}^+</math>.
 
As an example where the range equals the codomain, 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 (the function is a [[Surjective function|surjection]]), so the word range is unambiguous; it is the set of all real numbers.


==Formal definition==
==Formal definition==


Standard mathematical notation allows a formal definition of range.
When "range" is used to mean "codomain", the range of a function must be specified. It is often assumed to be the set of all real numbers, and {''y'' | there exists an ''x'' in the domain of ''f'' such that ''y'' = ''f''(''x'')} is called the image of ''f''.


In the first sense, the range of a function must be specified; it is often assumed to be the set of all real numbers, and {''y'' | there exists an ''x'' in the domain of ''f'' such that ''y'' = ''f''(''x'')} is called the image of ''f''.
When "range" is used to mean "image", the range of a function ''f'' is {''y'' | there exists an ''x'' in the domain of ''f'' such that ''y'' = ''f''(''x'')}.  In this case, the codomain of ''f'' must be specified, but is often assumed to be the set of all real numbers.
 
In the second sense, the range of a function ''f'' is {''y'' | there exists an ''x'' in the domain of ''f'' such that ''y'' = ''f''(''x'')}.  In this case, the codomain of ''f'' must be specified, but is often assumed to be the set of all real numbers.


In both cases, image ''f'' ⊆ range ''f'' ⊆ codomain ''f'', with at least one of the containments being equality.
In both cases, image ''f'' ⊆ range ''f'' ⊆ codomain ''f'', with at least one of the containments being equality.


==See also==
==See also==
*[[Bijection, injection and surjection]]
 
*[[Rescaled range]]
* [[Bijection, injection and surjection]]
*[[Range of a matrix]]
* [[Codomain]]
* [[Image (mathematics)]]
* [[Naive set theory]]
 
==Notes==
 
{{Reflist}}


==References==
==References==
{{reflist}}


{{logic}}
*{{Cite book
| first =
| last = 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
}}
*{{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
| title = Algebra
| publisher = Springer
| series = Graduate Texts in Mathematics
| volume = 73
| year = 1974
| isbn = 0-387-90518-9
| oclc = 703268
}}
*{{Cite book
| first = Walter
| last = Rudin
| title = Functional Analysis
| edition = 2nd
| publisher = McGraw Hill
| year = 1991
| isbn = 0-07-054236-8
}}
 
{{Mathematical logic}}


{{DEFAULTSORT:Range (Mathematics)}}
{{DEFAULTSORT:Range (Mathematics)}}

Latest revision as of 13:56, 8 November 2014

{{#invoke:Hatnote|hatnote}}

is a function from domain X to codomain Y. The smaller oval inside Y is the image of . Sometimes "range" refers to the image and sometimes to the codomain.

In mathematics, and more specifically in naive set theory, the range of a function refers to either the codomain or the image of the function, depending upon usage. Modern usage almost always uses range to mean image.

The codomain of a function is some arbitrary set. In real analysis, it is the real numbers. In complex analysis, it is the complex numbers.

The image of a function is the set of all outputs of the function. The image is always a subset of the codomain.

Distinguishing between the two uses

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.[1][2] More modern books, if they use the word "range" at all, generally use it to mean what is now called the image.[3] To avoid any confusion, a number of modern books don't use the word "range" at all.[4]

As an example of the two different usages, consider the function 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 , but its image is the set of non-negative real numbers , since is never negative if is real. For this function, if we use "range" to mean codomain, it refers to . When we use "range" to mean image, it refers to .

As an example where the range equals the codomain, consider the function , which inputs a real number and outputs its double. For this function, the codomain and the image are the same (the function is a surjection), so the word range is unambiguous; it is the set of all real numbers.

Formal definition

When "range" is used to mean "codomain", the range of a function must be specified. It is often assumed to be the set of all real numbers, and {y | there exists an x in the domain of f such that y = f(x)} is called the image of f.

When "range" is used to mean "image", the range of a function f is {y | there exists an x in the domain of f such that y = f(x)}. In this case, the codomain of f must be specified, but is often assumed to be the set of all real numbers.

In both cases, image f ⊆ range f ⊆ codomain f, with at least one of the containments being equality.

See also

Notes

  1. Hungerford 1974, page 3.
  2. Childs 1990, page 140.
  3. Dummit and Foote 2004, page 2.
  4. Rudin 1991, page 99.

References

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