Range (mathematics)

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For other uses, see Range.
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[edit]

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[edit]

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[edit]

Notes[edit]

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

References[edit]