
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 xaxis 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.
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 ≤ x ≤ 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 < 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 , 6 )
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 < x < 6
Note that the strict inequality sign (without equal) is used in the inequality notation of the domain because x =  4 and x = 6 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 < 2. The open 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 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 < 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 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 ≤ 4. The part on the right 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]
More Links and References
Find domain and range of functions,
Find the domain of a function ,
Step by Step Solver to Find the Domain of the Square Root of a Linear Function,
Find the Domain of the Square Root of a Quadratic Function 