Range

If  is a map (a.k.a. function, transformation, etc.) over a domain , then the range of , also called the image of  under , is defined as the set of all values that  can take as its argument varies over , i.e.,

Note that among mathematicians, the word "image" is used more commonly than "range."

The range is a subset of  and does not have to be all of .

Unfortunately, term "range" is often used to mean domain--its precise opposite--in probability theory, with Feller (1968, p. 200) and Evans et al. (2000, p. 5) calling the set of values that a variate  can assume (i.e., the set of values  that a probability density function  is defined over) the "range", denoted by  (Evans et al. 2000, p. 5).

Even worse, statistics most commonly uses "range" to refer to the completely different statistical quantity as the difference between the largest and smallest order statistics. In this work, this form of range is referred to as "statistical range."

REFERENCES:

Evans, M.; Hastings, N.; and Peacock, B. Statistical Distributions, 3rd ed. New York: Wiley, 2000.

Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd ed. New York: Wiley, 1968.