Range (mathematics)
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.^{}^{} More modern books, if they use the word "range" at all, generally use it to mean what is now called the image.^{} To avoid any confusion, a number of modern books don't use the word "range" at all.^{}
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 nonnegative 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
 Bijection, injection and surjection
 Codomain
 Image (mathematics)
 Naive set theory
Notes
References
 Childs (2009). A Concrete Introduction to Higher Algebra. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 9780387745275. OCLC 173498962.
 Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). Wiley. ISBN 9780471433347. OCLC 52559229.
 Hungerford, Thomas W. (1974). Algebra. Graduate Texts in Mathematics 73. Springer. ISBN 0387905189. OCLC 703268.
 Rudin, Walter (1991). Functional Analysis (2nd ed.). McGraw Hill. ISBN 0070542368.
