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

Given two sets X and Y, a binary relation f between X and Y is a (total) function (from X to Y) if for every x in X there is exactly one y in Y such that f relates x to y. The sets X and Y are called domain and codomain of f, respectively. The image of f is then the subset of Y consisting of only those elements y of Y such that there is at least one x in X with *f*(*x*) = *y*.

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]}

Given a function

with domain , the range of , sometimes denoted or ,^{[5]} may refer to the codomain or target set (i.e., the set into which all of the output of is constrained to fall), or to , the image of the domain of under (i.e., the subset of consisting of all actual outputs of ). The image of a function is always a subset of the codomain of the function.^{[6]}

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 ; if we use "range" to mean *image*, it refers to .

In many cases, the image and the codomain can coincide. For example, consider the function , 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.

**^**Hungerford 1974, page 3.**^**Childs 1990, page 140.**^**Dummit and Foote 2004, page 2.**^**Rudin 1991, page 99.**^**Weisstein, Eric W. "Range".*mathworld.wolfram.com*. Retrieved 2020-08-28.**^**Nykamp, Duane. "Range definition".*Math Insight*. Retrieved August 28, 2020.`{{cite web}}`

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- Childs (2009).
*A Concrete Introduction to Higher Algebra*. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 978-0-387-74527-5. OCLC 173498962. - Dummit, David S.; Foote, Richard M. (2004).
*Abstract Algebra*(3rd ed.). Wiley. ISBN 978-0-471-43334-7. OCLC 52559229. - Hungerford, Thomas W. (1974).
*Algebra*. Graduate Texts in Mathematics. Vol. 73. Springer. ISBN 0-387-90518-9. OCLC 703268. - Rudin, Walter (1991).
*Functional Analysis*(2nd ed.). McGraw Hill. ISBN 0-07-054236-8.