# 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 super-set of image. 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.^{[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]}

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
. 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 image of a function *f* is already implicitly defined. It is (by definition of image) the (maybe trivial) subset of the "range" which equals {*y* | there exists an *x* in the domain of *f* such that *y* = *f*(*x*)}.

When "range" is used to mean "image", the range of a function *f* is by definition {*y* | there exists an *x* in the domain of *f* such that *y* = *f*(*x*)}. In this case, the codomain of *f* must not be specified, because any codomain which contains this image as a (maybe trivial) subset will work.

In both cases, image *f* ⊆ range *f* ⊆ codomain *f*, with at least one of the containments being equality.

## See also

## Notes

## References

- 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.**73**. Springer. ISBN 0-387-90518-9. OCLC 703268. - Rudin, Walter (1991).
*Functional Analysis*(2nd ed.). McGraw Hill. ISBN 0-07-054236-8.