Range of a function

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

• The codomain of the function
• The image of the function

${\displaystyle f}$ 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.

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 ${\displaystyle f}$ may refer to the codomain or target set ${\displaystyle Y}$ (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 ${\displaystyle f}$ (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.

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 ${\displaystyle \mathbb {\displaystyle \mathbb {R} ^{}} }$. When we use "range" to mean image, it refers to ${\displaystyle \mathbb {R} ^{+}}$.

The image and the codomain can coincide. 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 (the function is a surjection), so the word range is unambiguous; it is the set of all real numbers.

See also

• Bijection, injection and surjection

Notes

1. Hungerford 1974, page 3.
2. Childs 1990, page 140.
3. Dummit and Foote 2004, page 2.
4. Rudin 1991, page 99.

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.