Range (mathematics)

Range as the image of a function

For a function f: X → Y the range (often written Range(f) or Im(f)) is the subset of Y consisting of those elements y for which there exists x in X with f(x)=y. The range is therefore the actual outputs of f, while the codomain Y is the set where outputs are allowed to lie. A function is surjective exactly when its range equals its codomain. Typical notation includes f(X) to denote the image of set X under f.

Range in statistics

In descriptive statistics the range of a data set is the difference between the largest and smallest observed values. Although easy to compute and interpret, it is sensitive to outliers and does not reflect the distribution's shape. It is often reported alongside other measures such as the interquartile range and standard deviation.

Range in linear algebra

For a linear transformation T: V → W between vector spaces the range is the subspace {T(v): v in V}. When a matrix represents T relative to bases, the range equals the column space of that matrix. Its dimension is the rank of T, and the rank–nullity theorem relates the rank (dimension of the range) to the dimension of the domain and kernel.

Computing and properties

• To find the range of a real-valued function one may solve y=f(x) for x or analyze limits and extrema for continuous functions.
• For matrices compute the span of columns to obtain the range; row-reduction reveals a basis and the rank.
• Under composition, Range(g◦f) ⊆ Range(g). Preimages differ: the preimage of a set under f lies in the domain, not the range.

Because the word "range" has multiple uses, writers often prefer the terms image, column space or data range to reduce ambiguity. For concise references and further reading see additional material.