Range of a function

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] More modern books, if they use the word "range" at all, generally use it to mean what is now called the image.[2] To avoid any confusion, a number of modern books don't use the word "range" at all.[3]

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 ${\displaystyle \operatorname {Range} (f)}$,[4] 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.[5]

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.

• Bijection, injection and surjection
• Essential range

• Childs, Lindsay N. (2009). Childs, Lindsay N. (ed.). A Concrete Introduction to Higher Algebra. Undergraduate Texts in Mathematics (3rd ed.). Springer. doi:10.1007/978-0-387-74725-5. 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. Vol. 73. Springer. doi:10.1007/978-1-4612-6101-8. ISBN 0-387-90518-9. OCLC 703268.
• Rudin, Walter (1991). Functional Analysis (2nd ed.). McGraw Hill. ISBN 0-07-054236-8.