Solutions to Questions on Domain and Range of Relations

Solutions to Questions on Domain and Range of Relations

Detailed solutions and explanations to questions on how find the domain and range of a relation are presented.

For each relation below given by its graph, find the domain and range and state whether the relation is a function


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

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