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Range of a function

## Summary

${\displaystyle f}$ is a function from domain X to codomain Y. The yellow oval inside Y is the image of ${\displaystyle f}$. Sometimes "range" refers to the image and sometimes to the codomain.

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.

## Terminology

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]

## Elaboration and example

Given a function

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

with domain ${\displaystyle X}$, the range of ${\displaystyle f}$, sometimes denoted ${\displaystyle \operatorname {ran} (f)}$ or ${\displaystyle \operatorname {Range} (f)}$,[5] 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.[6]

As an example of the two different usages, consider the function ${\displaystyle f(x)=x^{2}}$ 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 ${\displaystyle \mathbb {R} }$, 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 mean codomain, it refers to ${\displaystyle \mathbb {\displaystyle \mathbb {R} ^{}} }$; if we use "range" to mean image, it refers to ${\displaystyle \mathbb {R} ^{+}}$.

In many cases, the image and the codomain can coincide. For example, consider the function ${\displaystyle f(x)=2x}$, 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.

## Notes and References

1. ^ Hungerford 1974, page 3.
2. ^ Childs 1990, page 140.
3. ^ Dummit and Foote 2004, page 2.
4. ^ Rudin 1991, page 99.
5. ^ Weisstein, Eric W. "Range". mathworld.wolfram.com. Retrieved 2020-08-28.
6. ^ Nykamp, Duane. "Range definition". Math Insight. Retrieved August 28, 2020.

## Bibliography

• 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.