Domain and range

Functions. A function $f$ is a ``correspondence'' from each element $x$ of a set $ D$ to ``exactly'' one element $y$ of a set $ E$. The domain of $f$ is always the set $ D$ itself, but the range of $f$ must be defined as the subset $ R$ of $ E$ consisting of all possible values $f(x)$ for $x$ in $ D$. In mathematical terms the range $ R$ of $f$ can be expressed as

$\displaystyle R = \{f(x): x \in D\}.

For example, both the domain and the range of a linear function $ f(x) = a x + b$ are given by the set of entire real numbers. In other words, $ D = R = \mathbb{R}$.

Intervals. An interval is a set of real numbers ``between two endpoints,'' say $a$ and $b$. Depending on whether it includes endpoints or not, and whether it is finite or infinite, the following types of an interval are considered.

Finite interval Notation
$ a < x < b$ $ (a,\: b)$
$ a \le x \le b$ $ [a,\: b]$
$ a \le x < b$ $ [a,\: b)$
$ a < x \le b$ $ (a,\: b]$

Infinite interval Notation
$ x > a$ $ (a,\: \infty)$
$ x \ge a$ $ [a,\: \infty)$
$ x < b$ $ (-\infty,\: b)$
$ x \le b$ $ (-\infty,\: b]$

The set of all real numbers can be viewed as a special case of interval, denoted by  $ (-\infty, \infty)$.

Department of Mathematics
Last modified: 2005-09-29