Range of a function

f is a function from domain X to codomain Y. The smaller oval inside Y is the range of f.

In mathematics, the range of a function $f$ (denoted by ${\mbox{ran}}\,f$ ) is the set of all "output" values produced by f.

Sometimes it is called the image, or more precisely, the image of the domain of the function. If a function is a surjection then its range is equal to its codomain. In a representation of a function in an xy Cartesian coordinate system, the range is represented on the ordinate (on the y axis). Sometimes the range means the set of all possible values – the codomain. This is also the current usage for range in computer science. The ambiguity makes it preferable to specify whether it is the image or the codomain being discussed.

Many ranges are of the form of an interval, e.g. $[0,1)$ for the numbers from 0 to 1 including 0 and excluding 1. Thus intervals are often referred to as ranges.

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 multiplies it by two. Multiplication by any real number but 0 can yield any real number, therefore f(x) may be any real number, which is to say that the range of f is (−∞, ∞).

In other instances, range may be restricted by the domain of the function. Consider the function g such that $g\colon \mathbb {R} ^{+}\rightarrow \mathbb {R} ,g(x)=2x$ . Since the codomain is $\mathbb {R}$ , any real number is a legal value for g(x). However, the domain of g is $\mathbb {R} ^{+}$ , so the input for g may only be a real number greater than zero. Multiplying any positive real number by two will always yield another positive real number, so the range of g is (0, ∞). Note here that the range of the function is not equal to its codomain, though the range is (and always will be) a subset of the codomain.

Range may also be restricted by the definition of the function. Consider the function h such that $h\colon \mathbb {R} \rightarrow \mathbb {R} ,h(x)=x^{2}$ . Here the input is again any real number, though squaring any real number will never yield a negative number, and so the output of h may be any nonnegative number (including zero), thus the range is [0, ∞).

Examples

See also