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

In some cases the codomain and the image of a function are the same set; such a function is called *surjective* or *onto*. For any non-surjective function the codomain and the image are different; however, a new function can be defined with the original function's image as its codomain, where This new function is surjective.

Given two sets X and Y, a binary relation f between X and Y is a function (from X to Y) if for every element x in X there is exactly one y in Y such that f relates x to y. The sets X and Y are called the *domain* and *codomain* of f, respectively. The *image* of the function f is 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]} More modern books, if they use the word "range" at all, generally use it to mean what is now called the image.^{[2]} To avoid any confusion, a number of modern books don't use the word "range" at all.^{[3]}

Given a function

with domain , the range of , sometimes denoted or ,^{[4]} 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.^{[5]}

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 .

For some functions, the image and the codomain coincide; these functions are called *surjective* or *onto*. For example, consider the function 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 word *range* is 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 the integers to the integers, the doubling function is not surjective because only the even integers are part of the image. However, a new function whose domain is the integers and whose codomain is the even integers *is* surjective. For the word *range* is unambiguous.

**^**Hungerford 1974, p. 3; Childs 2009, p. 140.**^**Dummit & Foote 2004, p. 2.**^**Rudin 1991, p. 99.**^**Weisstein, Eric W. "Range".*mathworld.wolfram.com*. Retrieved 2020-08-28.**^**Nykamp, Duane. "Range definition".*Math Insight*. Retrieved August 28, 2020.

- 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. 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. doi:10.1007/978-1-4612-6101-8. ISBN 0-387-90518-9. OCLC 703268. - Rudin, Walter (1991).
*Functional Analysis*(2nd ed.). McGraw Hill. ISBN 0-07-054236-8.

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