How to find the Domain and Range of a relation given by its graphs?

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 a) 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 domain and range of the relation given by its graph shown below and 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 domain and range of the relation given by its graph shown below and 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 domain and range of the relation given by its graph shown below and 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.

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

Detailed Solutions and explanations to these questions.

a)

b)

c)

d)

e)

Detailed Solutions and explanations to these questions.