# Find Domain and Range of Relations Given by Graphs Examples and Questions With Solutions

Find the domain and range of a relation given by its graph. Examples are presented along with detailed Solutions and explanations and also more questions with detailed solutions.

## Examples with Solutions

Example 1
a) Find the domain and b) the range of the relation given by its graph shown below and c) state whether the relation is a function or not.

Solution:
a)
Domain: We first find the 2 points on the graph of the given relation with the smallest and the largest x-coordinate. In this example the 2 points are A(-2,-4) and B(4,-6) (see graph above). The domain is the set of all x values from the smallest x-coordinate (that of A) to the largest x-coordinate (that of B) and is written as:
-2 ≤ x ≤ 4
The double inequality above has the inequality symbol ≤ at both sides because the closed circles at points A and B indicate that the relation is defined at these values of x.
b)
Range: We need to find the coordinates of the 2 points on the graph with the lowest and the largest values of the y coordinate. In this example, these points are B(4,-6) and C(2,2). The range is the set of all y values between the smallest and the largest y coordinates and given by the double inequality:
-6 ≤ y ≤ 2
The inequality symbol ≤ is used at both sides because the closed circles at points B and C indicates the relation is defined at these values.
c)
The relation graphed above is a function because no vertical line can intersect the given graph at more than one point.

Example 2
Find the a) domain and b) range of the relation given by its graph shown below and c) state whether the relation is a function or not.

Solution:
a)
Domain: In this example points A(-3,-5) and B(8,4) have the smallest and the largest x-coordinates respectively, hence the domain is given by:
-3 ≤ x ≤ 8
The use of the symbol ≤ at both sides is due to the fact that the relation is defined at points A and B (closed circles at both points).
b)
Range: Points A and B have the smallest and the largest values of the y-coordinate respectively. The range is given by the inequality:
- 5≤ y ≤ 4
The use of the symbol ≤ at both sides is due to the fact that the relation is defined at points A and B.
c)
No vertical line can cut the given graph at more than one point and therefore the relation graphed above is a function.

Example 3
Find the a) domain and b) range of the relation given by its graph shown below and c) state whether the relation is a function or not.

Solution:
a)
Domain: Points A(-3,-2) and B(1,-2) have the smallest and the largest x-coordinates respectively, hence the domain:
-3 ≤ x ≤ 1
The use of the symbol ≤ at both sides is due to the fact that the relation is defined at points A and B (closed circles at both points).
b)
Range: Points C(-1,-5) and D(-1,1) have the smallest and the largest y-coordinate respectively. The range is given by the double inequality:
- 5≤ y ≤ 1
The relation is defined at points C and D (closed circles), hence the use of the inequality symbol ≤.
c)
There is at least one vertical line that cuts the given graph at two points (see graph below) and therefore the relation graphed above is NOT a function.

Example 4
Find the a) domain and b) range of the relation given by its graph shown below and c) state whether the relation is a function or not.

Solution:
a)
Domain: Points A(-3,0) has the smallest x-coordinate. The arrow at the top right of the graph indicates that the graph continues to the left as x increases. Hence there is no limit to the largest x-coordinate of points on the graph. The domain is given by all values greater than or equal to the smallest values x = -3 and is written as:
x ≥ -3
The use of the symbol ≥ at because the relation is defined at points A (closed circle at point A).
b)
Range: Points B and C have equal and smallest y-coordinates equal to -2. The arrow at the top right of the graph indicates that the y coordinate increases as x increases. Hence there is no limit to the y-coordinate and therefore the range is given by all values greater than or equal to the smallest value y = -2 and is written as:
y ≥ -2
The use of the inequality symbol ≥ is due to the fact that the relation is defined at y = -2 (closed circle at B and C).
c)
There is no vertical line that cuts the given graph at more than one point (see graph below) and therefore the relation graphed above is a function.

Example 5
Find the a) domain and b) range of the relation given by its graph shown below and c) state whether the relation is a function or not.

Solution:
a)
Domain: Points A(-2,-3) has the smallest x-coordinate. The arrow at the top right of the graph indicates that the graph continues to the left as x increases. Hence there is no limit to the largest x-coordinate of points on the graph. The domain is given by all values greater than the smallest values x = - 2 and is written as:
x > -2
We use of the inequality symbol > (with no equal) because the relation is not defined at points A (open circle at point A).
b)
Range: Points A(-2,-3) has the smallest y-coordinate equal to - 3. The arrow at the top right of the graph indicates that the y coordinate increases as x increases. Therefore there is no limit to the y-coordinate. Hence the range is given by all values greater than the smallest value y = - 3 and is written as:
y > - 3
The inequality symbol > is used because the relation is not defined at y = - 3 (open circle at point A).
c)
The graph represents a function because there is no vertical line that cuts the given graph at more than one point.

## More Questions

and their detailed solutions

For each relation below, find the domain and range and state whether the relation is a function.
1)

2)

3

4)

5)

## More References and links

Domain and Range of Functions
Online tutorials on how to find the domain and range of functions.
Middle School Maths (Grades 6, 7, 8, 9) - Free Questions and Problems With Answers
High School Maths (Grades 10, 11 and 12) - Free Questions and Problems With Answers

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