Inmathematics, therange of a function may refer either to thecodomain of thefunction, or theimage of the function. In some cases the codomain and the image of a function are the same set; such a function is calledsurjective oronto. For any non-surjective function${\displaystyle f:X\to Y,}$ the codomain${\displaystyle Y}$ and the image${\displaystyle \tilde{Y}}$ are different; however, a new function can be defined with the original function's image as its codomain,${\displaystyle \tilde{f}:X\to \tilde{Y}}$ where${\displaystyle \tilde{f}(x)=f(x).}$ This new function is surjective.

Given twosetsX andY, abinary relationf betweenX andY is a function (fromX toY) if for everyelementx inX there is exactly oney inY such thatf relatesx toy. The setsX andY are called thedomain andcodomain off, respectively. Theimage of the functionf is thesubset ofY consisting of only those elementsy ofY such that there is at least onex inX withf(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 thecodomain.^[1] More modern books, if they use the word "range" at all, generally use it to mean what is now called theimage.^[2] To avoid any confusion, a number of modern books don't use the word "range" at all.^[3]

Given a function

$${\displaystyle f:X\to Y}$$

withdomain${\displaystyle X}$, the range of${\displaystyle f}$, sometimes denoted${\displaystyle \mathrm{ran}(f)}$ or${\displaystyle \mathrm{Range}(f)}$,^[4] may refer to the codomain or target set${\displaystyle Y}$ (i.e., the set into which all of the output of${\displaystyle f}$ is constrained to fall), or to${\displaystyle f(X)}$, the image of the domain of${\displaystyle f}$ under${\displaystyle f}$ (i.e., the subset of${\displaystyle Y}$ consisting of all actual outputs of${\displaystyle f}$). 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${\displaystyle f(x)=x^2}$ as it is used inreal analysis (that is, as a function that inputs areal number and outputs its square). In this case, its codomain is the set of real numbers${\displaystyle \mathbb{R}}$, but its image is the set of non-negative real numbers${\displaystyle {\mathbb{R}}^{+}}$, since${\displaystyle x^2}$ is never negative if${\displaystyle x}$ is real. For this function, if we use "range" to meancodomain, it refers to${\displaystyle {\displaystyle {\mathbb{R}}^{}}}$; if we use "range" to meanimage, it refers to${\displaystyle {\mathbb{R}}^{+}}$.

For some functions, the image and the codomain coincide; these functions are calledsurjective oronto. For example, consider the function${\displaystyle f(x)=2x,}$ 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 wordrange 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 theintegers to the integers, the doubling function${\displaystyle f(n)=2n}$ is not surjective because only theeven integers are part of the image. However, a new function${\displaystyle \tilde{f}(n)=2n}$ whose domain is the integers and whose codomain is the even integersis surjective. For${\displaystyle \tilde{f},}$ the wordrange is unambiguous.

• Bijection, injection and surjection
• Essential range

• 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.

• Functions and mappings
• Basic concepts in set theory

• Articles with short description
• Short description is different from Wikidata