Range (mathematics)

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

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.^[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 $f(x) = x^2$ 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 $\mathbb{R}$, but its image is the set of non-negative real numbers $\mathbb{R}^+$, since $x^2$ is never negative if $x$ is real. For this function, if we use "range" to mean codomain, it refers to $\mathbb{R}$. When we use "range" to mean image, it refers to $\mathbb{R}^+$.

As an example where the range equals the codomain, consider the function $f(x) = 2x$, 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.

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

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.