Domain and Range

A relation may be described by:
  • a listed set of ordered pairs
  • a graph
  • a rule
The set of all first elements of a set of ordered pairs is known as the domain and the set of all second elements of a set of ordered pairs is known as the range.
Alternatively, the domain is the set of independent values and the range is the set of dependent values.

The domain of a relation is the set of values of $x$ in the relation.
The range of a relation is the set of values of $y$ in the relation.

The domain and range of a relation are often described using set notation.

If a relation id described by a rule, it should also specify the domain. For example:
  • the relation $\{(x,y):y=x+2,x\in\{1,2,3\}\}$ describes the set of ordered pairs $\{(1,3),(2,4),(3,5)\}$
  • the domain is the set $X=\{1,2,3\}$, where is given
  • the range is the set $Y=\{3,4,5\}$, and can be found by applying the rule $y=x+2$ to the domain values

If the domain of a relation is not specifically stated, it is assumed to consist of all real numbers for which the rule has meaning. This is referred to as the implied domain of a relation.
  • $y=x^2$ has the implied domain $x \in \mathbb{R}$, and implied range $y=x^2 \ge 0$, where $y \in \mathbb{R}$.
  • $y=\sqrt{x}$ has the implied domain $x \ge 0$, where $x \in \mathbb{R}$, and implied range $y=\sqrt{x} \ge 0$, where $y \in \mathbb{R}$.
  • $y=\dfrac{1}{x}$ has the implied domain $x \ne 0$, where $x \in \mathbb{R}$, and implied range $y=\dfrac{1}{x} \ge 0$, where $y \in \mathbb{R}$.
  • $y=\dfrac{1}{\sqrt{x}}$ has the implied domain $x \gt 0$, where $x \in \mathbb{R}$, and implied range $y=\dfrac{1}{x} \gt 0$, where $y \in \mathbb{R}$.

Example 1

State the domain and range of $\{(2,4),(3,9),(4,14),(5,19)\}$.

Example 2

State the domain and range of the graph. Domain and Range

Example 3

State the domain and range of the graph.

Example 4

State the domain and range of $y=\sqrt{x}$.

Example 5

State the domain and range of $y=\dfrac{1}{x+1}$.