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

Find the domain and range of a relation given by its graph. Questions 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.

graph of relation for example 1


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

graph of relation for example 2


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 domain and range of the relation given by its graph shown below and state whether the relation is a function or not.

graph of relation for example 3


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.

graph of relation for example 3


Example 4

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

graph of relation for example 4


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 domain and range of the relation given by its graph shown below and state whether the relation is a function or not.

graph of relation for example 5


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

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

graph of relation for question 1


b)

graph of relation for question 2


c)

graph of relation for question 3


d)

graph of relation for question 4


e)

graph of relation for question 5


Solutions to the Above Questions

a)

graph of relation for question 1


Solution


a) Domain: Points A(-8 , - 0.5) and B(4,0) have the smallest and the largest x-coordinate respectively. The domain is the set of all x values between the smallest x-coordinate (that of A) to the largest x-coordinate (that of B) and is written as:
- 8 ≤ x ≤ 4
Since the relation is defined at both points (closed circle) the inequality symbol ≤ is used.
b) Range: Points C(-3,-5) and B(4,0) have the smallest and largest y-coordinates respectively. Hence, the range is the set of all y values between the smallest and the largest y coordinates and given by the double inequality:
- 5 ≤ y ≤ 0
The inequality symbol ≤ is used at both sides the relation is defined at these y values (closed circles).
c) The relation graphed above is a function because no vertical line can intersect the given graph at more than one point.



b)

graph of relation for question 2


Solution


a) Domain: Points A(-2 , 4) and B(4,6) have the smallest and the largest x-coordinate respectively. 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
Closed circles at both point A and B hence the use of the inequality symbol ≤.
b) Range: Points C(2,-2) and B(4,6) have the smallest and largest y-coordinates respectively. Hence, the range is the set of all y values between the smallest and the largest y coordinates and given by the double inequality:
- 2 ≤ y ≤ 6
The inequality symbol ≤ is used at both sides the relation is defined at these y values (closed circle).
c) The relation graphed above is a function because no vertical line can intersect the given graph at more than one point.



c)

graph of relation for question 3


Solution


a) Domain: Point A(4 , 2) has the largest x-coordinate. As x decreases (moving left), the arrow at the top left indicates that there is no limit to the smallest value of the x-coordinate of any point on the given graph. The domain is the set of all x values smaller than 4 and is written as:
x ≤ 4
The closed circle at point A means the relation is defined at x = 4, hence use of the inequality symbol ≤.
b) Range: Points B(2,-2) and C(-2,-2) have the smallest (and equal) y-coordinates. The arrow on the top left indicates that as x decreases (moving left), the y coordinate of points on the graph increases without limit. Hence, the range is the set of all y values greater than or equal to -2 and is given by the inequality:
y ≥ -2
The inequality symbol ≤ is used because the relation is defined at y = 4 (closed circle).
c) The relation graphed above is a function because no vertical line can intersect the given graph at more than one point.



d)

graph of relation for question 4


Solution


a) Domain: Points A(-5 , -1) and B(1, -1) have the smallest and the largest x-coordinate respectively. The domain is the set of all x values between - 5 and - 1 and is given by:
- 5 ≤ x ≤ 1
The closed circle at points A and B means the relation is defined at x = - 5 and x = 1, hence use of the inequality symbol ≤ at both sides.
b) Range: Points C(-2,-3) and D(-2,1) have the smallest and the largest y-coordinates respectively. Hence, the range is the set of all y values between -3 and 1 and is given by:
-3 ≤ y ≤ 1
The inequality symbol ≤ is used because the relation is defined at both points (closed circle).
c) The relation graphed above is NOT a function because at least one vertical line intersects the given graph at two points as shown below.

graph of relation for question 4



e)

graph of relation for question 5


Solution


a) Domain: Point A(-3 , 1.8) has the smallest x-coordinate. As x increases (moving right), the arrow at the bottom right, indicates that there is no limit to the largest value of the x-coordinate of any point on the given graph. The domain is the set of all x values greater than -3 and is written as:
x > -3
The open circle at point A means the relation is not defined at x = -3, hence use of the inequality symbol >.
b) Range: Points B(-2,2) have the largest y-coordinates. The arrow on the bottom right indicates that as x increases (moving right), the y coordinate of points on the graph decreases without limit. Hence, the range is the set of all y values smaller than or equal to 2 and is given by the inequality:
y ≤ 2
The inequality symbol ≤ is used because the relation is defined at y = 2 (closed circle at B).
c) The relation graphed above is a function because no vertical line can intersect the given graph at more than one point.


More References and links

Domain and Range of Functions
Online tutorials on how to find the domain and range of functions.
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