# Domain and Range – An Ultimate Guide

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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