# Domain of a Graph

## Domain of a Graph of a Function

The imply world of a serve degree fahrenheit is the set of all values of ten for which farad ( x ) is defined and real. The graph of a function fluorine is the set of all points ( adam, f ( x ) ). Hence, For a affair degree fahrenheit defined by its graph, the imply domain of degree fahrenheit is the specify of all the real values x along the x-axis for which there is a point on the given graph.
As an modelthere 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 ten are included in the world of fluorine.
There are no points on the graph at x = – 1 ( open r-2 on the graph ), 0.5, 1, 1.5, 2 ( overt circle ). These values and more early values of x are not included in the knowledge domain.

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

## Domain of a Graph; Examples with Detailed Solutions

### Example 1

Find the world 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 ten = 6. For all x between -4 and 6, there points on the graph. Hence the world, in interval notation, is written as
[ -4, 6 ]
In inequality note, the world is written as
– 4 ≤ x ≤ 6
Note that we close the brackets of the interval because -4 and 6 are included in the sphere which is indicated by the closed circles at x = – 4 and x = 6.

### Example 2

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

Solution to Example 2
The graph starts at x = – 4 and ends ten = 4. There are points on the graph for all values of x between – 4 and 4 including at – 4 and 4. Hence the knowledge 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 ten = 8. The graph is defined for all x between – 8 and 8. We include – 8 and 8 because of the shut circles at x = – 8 and x = 8. Hence the knowledge domain, in time interval notation, is written as
[ – 8, 8 ]

### Example 4

Find the world of the graph of the affair shown below.

Solution to Example 4
The graph starts at x = – 4 and ends ten = 6. The graph is defined for all x between – 4 and 6. The interval is closed at – 4 and 6 because of the close circles at x = – 4 and x = 6. Hence the sphere, 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 < 4. x = - 4 and x = 4 are not included in the world because of the open circles at these values. Hence the domain, in interval notation, is written as
( – 4, 6 )
note that the interval is open to indicate that – 4 and 4 are not included in the sphere of the graph.
In inequality notation, the lapp domain is given by
– 4 < x < 6
Note that the rigorous inequality sign ( without peer ) is used in the inequality notation of the world because x = – 4 and x = 6 are not included in the knowledge domain.

### Example 6

Write the world of the graph of the serve shown below in interval note.

Solution to Example 6
The graph starts at values of x = – 8 and ends at values of x < 2. The candid circles at x = - 4, x = -2 and x = 2 indicates that these values are not included in the domain. Hence the domain, in interval notation, is written as
[ – 8, – 4 ) ∪ ( – 4, -2 ) ∪ ( -2, 2 )
Note the interval is candid 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 note.

Solution to Example 7
The graph starts at x = – 4 and ends adam < 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 ≤ x < 2

### Example 8

Write the sphere of the graph of the function shown below using time interval note

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