Range of a function

For the statistical concept, seeRange (statistics).

$f$is a function fromdomainXtocodomainY.The yellow oval insideYis theimageof$f$.Sometimes "range" refers to the image and sometimes to the codomain.

Inmathematics,therange of a functionmay refer to either of two closely related concepts:

• thecodomainof thefunction,or
• theimageof the function.

In some cases the codomain and the image of a function are the same set; such a function is calledsurjectiveoronto.For any non-surjective function$f:X\to Y,$the codomain$Y$and the image${\tilde {Y}}$are different; however, a new function can be defined with the original function's image as its codomain,${\tilde {f}}:X\to {\tilde {Y}}$where${\tilde {f}}(x)=f(x).$This new function is surjective.

Definitions

Given twosetsXandY,abinary relationfbetweenXandYis a function (fromXtoY) if for everyelementxinXthere is exactly oneyinYsuch thatfrelatesxtoy.The setsXandYare called thedomainandcodomainoff,respectively. Theimageof the functionfis thesubsetofYconsisting of only those elementsyofYsuch that there is at least onexinXwithf(x) =y.

Usage

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 thecodomain.More modern books, if they use the word "range" at all, generally use it to mean what is now called theimage.To avoid any confusion, a number of modern books don't use the word "range" at all.

Elaboration and example

Given a function

$f\colon X\to Y$

withdomain$X$,the range of$f$,sometimes denoted$\operatorname {ran} (f)$or$\operatorname {Range} (f)$,may refer to the codomain or target set$Y$(i.e., the set into which all of the output of$f$is constrained to fall), or to$f(X)$,the image of the domain of$f$under$f$(i.e., the subset of$Y$consisting of all actual outputs of$f$). The image of a function is always a subset of the codomain of the function.

As an example of the two different usages, consider the function$f(x)=x^{2}$as it is used inreal analysis(that is, as a function that inputs areal numberand outputs its square). In this case, its codomain is the set of real numbers$\mathbb {R}$,but its image is the set of non-negative real numbers$\mathbb {R} ^{+}$,since$x^{2}$is never negative if$x$is real. For this function, if we use "range" to meancodomain,it refers to$\mathbb {\displaystyle \mathbb {R} ^{}}$;if we use "range" to meanimage,it refers to$\mathbb {R} ^{+}$.

For some functions, the image and the codomain coincide; these functions are calledsurjectiveoronto.For example, consider the function$f(x)=2x,$which inputs a real number and outputs its double. For this function, both the codomain and the image are the set of all real numbers, so the wordrangeis unambiguous.

Even in cases where the image and codomain of a function are different, a new function can be uniquely defined with its codomain as the image of the original function. For example, as a function from theintegersto the integers, the doubling function$f(n)=2n$is not surjective because only theeven integersare part of the image. However, a new function${\tilde {f}}(n)=2n$whose domain is the integers and whose codomain is the even integersissurjective. For${\tilde {f}},$the wordrangeis unambiguous.

See also

• Bijection, injection and surjection
• Essential range

Bibliography

• Childs, Lindsay N. (2009). Childs, Lindsay N. (ed.).A Concrete Introduction to Higher Algebra.Undergraduate Texts in Mathematics(3rd ed.). Springer.doi:10.1007/978-0-387-74725-5.ISBN978-0-387-74527-5.OCLC173498962.
• Dummit, David S.; Foote, Richard M. (2004).Abstract Algebra(3rd ed.). Wiley.ISBN978-0-471-43334-7.OCLC52559229.
• Hungerford, Thomas W.(1974).Algebra.Graduate Texts in Mathematics.Vol. 73. Springer.doi:10.1007/978-1-4612-6101-8.ISBN0-387-90518-9.OCLC703268.
• Rudin, Walter (1991).Functional Analysis(2nd ed.). McGraw Hill.ISBN0-07-054236-8.