# Domain of a Graph

Find the domain of a graph of a function; examples with solutions are presented. First the graphical meaning of the concept of the domain of a function is explained.

## Domain of a Graph of a Function

The implied domain of a function $$f$$ is the set of all values of $$x$$ for which $$f(x)$$ is defined and real. The graph of a function $$f$$ is the set of all points $$(x, f(x))$$. Hence, For a function $$f$$ defined by its graph, the implied domain of $$f$$ is the set of all the real values $$x$$ along the $$x$$-axis for which there is a point on the given graph.
As an example there are points on the graph below at $$x = -3, -2.5, -2, -0.5, 2.5, 3, 3.2, 4$$. These values and more other values of $$x$$ are included in the domain of $$f$$.
There are no points on the graph at $$x = -1$$ (open circle on the graph), $$0.5$$, $$1$$, $$1.5$$, $$2$$ (open circle). These values and more other values of $$x$$ are not included in the domain of $$f$$.

With these ideas and definition, we will now solve examples where the whole domain of a given graph is found.

## Domain of a Graph; Examples with Detailed Solutions

### Example 1

Find the domain of the graph of the function shown below and write it in both interval and inequality notations.

Solution to Example 1
The graph starts at $$x = -4$$ and ends $$x = 6$$. For all $$x$$ between $$-4$$ and $$6$$, there points on the graph. Hence the domain, in interval notation, is written as
$[-4 , 6]$
In inequality notation, the domain is written as
$-4 \leq x \leq 6$
Note that we close the brackets of the interval because $$-4$$ and $$6$$ are included in the domain which is indicated by the closed circles at $$x = -4$$ and $$x = 6$$.

### Example 2

What is the domain, in interval notation, of the graph of the function shown below?

Solution to Example 2
The graph starts at $$x = -4$$ and ends $$x = 4$$. There are points on the graph for all values of $$x$$ between $$-4$$ and $$4$$ including at $$-4$$ and $$4$$. Hence the domain, in interval notation, is written as
$[-4, 4]$

### Example 3

What is the domain of the graph of the function?

Solution to Example 3
The graph starts at $$x = -8$$ and ends $$x = 8$$. The graph is defined for all $$x$$ between $$-8$$ and $$8$$. We include $$-8$$ and $$8$$ because of the closed circles at $$x = -8$$ and $$x = 8$$. Hence the domain, in interval notation, is written as
$[-8, 8]$

### Example 4

Find the domain of the graph of the function shown below.

Solution to Example 4
The graph starts at $$x = -4$$ and ends $$x = 6$$. The graph is defined for all $$x$$ between $$-4$$ and $$6$$. The interval is closed at $$-4$$ and $$6$$ because of the closed circles at $$x = -4$$ and $$x = 6$$. Hence the domain, in interval notation is written as
$[-4, 6]$

### Example 5

Write the domain of the graph of the function shown below in interval and inequality notations.

Solution to Example 5
The graph starts at values of $$x > -4$$ and ends at values of $$x \lt 4$$. $$x = -4$$ and $$x = 4$$ are not included in the domain because of the open circles at these values. Hence the domain, in interval notation, is written as
$(-4 , 4)$
Note that the interval is open to indicate that $$-4$$ and $$4$$ are not included in the domain of the graph.
In inequality notation, the same domain is given by
$-4 \lt x \lt 4$
Note that the strict inequality sign (without equal) is used in the inequality notation of the domain because $$x = -4$$ and $$x = 4$$ are not included in the domain.

### Example 6

Write the domain of the graph of the function shown below in interval notation.

Solution to Example 6
The graph starts at values of $$x = -8$$ and ends at values of $$x \lt 2$$. The open circles at $$x = -4$$, $$x = -2$$, and $$x = 2$$ indicate that these values are not included in the domain. Hence the domain, in interval notation, is written as
$[-8, -4) \cup (-4, -2) \cup (-2, 2)$
Note the interval is open at $$x = -4$$, $$x = -2$$, and $$x = 2$$ to indicate that these values are not included in the domain of the graph.

### Example 7

Write the domain of the graph of the function shown below in inequality notation.

Solution to Example 7
The graph starts at $$x = -4$$ and ends $$x \lt 2$$. The domain does not include $$x = 2$$ because of the open circle at $$x = 2$$. Hence the domain, in inequality notation, is written as
$-4 \leq x \lt 2$

In interval notation, the domain is given by $[-4 , 2)$

### Example 8

Write the domain of the graph of the function shown below using interval notation

Solution to Example 8
The graph is made up of three parts. The left part is defined for all values of $$x$$ between $$-4$$ and $$-2$$. The part in the center is defined on the interval $$x > 0$$ and $$x \leq 4$$. The part on the right is defined for $$x > 6$$ and $$x \leq 8$$. The domain is written as a union of three intervals as follows
$[-4, -2] \cup (0, 4] \cup (6, 8]$