Range of a function

This article is about range of a function. For range in statistics, see Range (statistics).

f is a function from domain X to codomain Y. The smaller oval inside Y is sometimes called the "range" of f, sometimes the "image" of f.

In mathematics, the range of a function refers to the output of a function, but there is not universal agreement on the subject of whether the output is the range or is included in the range. This disagreement among mathematicans is illustrated by the function $f$ with $f(x)=x^{2}$. Some books say that both the domain and range of this function is the set of all real numbers. These books call the actual output of the function the image. This is the current usage for range in computer science. Other books say that the range is the set of non-negative real numbers, since every output is non-negative. In this case, the larger set containing the range is called the co-domain. This usage is more common in modern mathematics.

Because of this ambiguity, it is a good idea to specify whether it is the image or the co-domain being discussed.

Examples

Let f be a function on the real numbers $f\colon \mathbb {R} \rightarrow \mathbb {R}$ defined by $f(x)=2x$. This function takes as input any real number and outputs a real number two times the input. In this case, the range is unambiguous; it is the set of all real numbers.

In contrast, consider the function $f(x)=sin(x)$. If the word "range" is used in the first sense given above, we would say the range of f is all real numbers, but since the output of the sine function is always between -1 and 1, "range" in the second sense would say the range is the closed interval from -1 to 1.

Formal definition

Standard mathematical notation allows a formal definition of range.

In the first sense, 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 the second sense, the range of a function must be specified, 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.

In both cases, image f $\subseteq$ range f $\subseteq$ codomain f, with one or the other containment being equality.

See also

• Bijection, injection and surjection
• Rescaled range
• Range of a matrix