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 rule describes a relation, 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 $\{(2,4),(3,9),(4,14),(5,19)\}$.
Domain: $\{2,3,4,5\}$
Range: $\{4,9,14,19\}$
Example 2
State the domain and range of the graph.
Domain: $x \in \mathbb{R}$
Range: $y \ge 0, y \in \mathbb{R}$
Example 3
State the domain and range of the graph.
Domain: $-3 \le x \le 3$
Range: $-3 \le y \le 3$
Example 4
State the domain and range of $y=\sqrt{x}$.
Domain: $x \ge 0, x \in \mathbb{R}$
Range: $y \ge 0, y \in \mathbb{R}$
Example 5
State the domain and range of $y=\dfrac{1}{x+1}$.
Domain: $x \ne -1, x \in \mathbb{R}$
Range: $y \ne 0, y \in \mathbb{R}$
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