9.2 Relations

CORD MathCORD Math

Mrs. SpitzMrs. Spitz

Fall 2006Fall 2006

Objectives

• Identify the domain, range and inverse of a relation, and

• Show relations as sets of ordered pairs and mappings

Assignment

• Pgs 361-363 #4-41

Definition of the Domain and Range of a Relation

• The domain of a relation is the set of all first coordinates from the ordered pairs. The range of the relation is the set of all second coordinates from the ordered pairs.

Table/Mapping/Graph – What’s the difference?

• A relation can also be shown using a table, mapping or a graph. A mapping illustrates how each element of the domain is paired with an element in the range. For example, the relation {(2, 2), (-2, 3),(0, -1)} can be shown in each of the following ways.

x y

2 2

-2 3

0 -1

2

-2

0

2

3

-1

x y

Ex. 1: Express the relation shown in the table below as a set of ordered pairs. Then determine the domain and range.

x y

0 5

2 3

1 -4

-3 3

-1 -2

Ordered Pairs:

(0, 5), (2, 3), (1, -4), (-3, 3), and (-1, -2)

Domain (all x values):

{0, 2, 1, -3, -1}

Range (all y values):

{5, 3, -4, 3, -2}

Ex. 2: Express the relation shown in the graph below as a set of ordered pairs. Then determine the domain and range. Then show the relation using a mapping.

Ordered Pairs:

(-4, -2), (-2, 1), (0, 2), (1, -3), and (3, 1)

Domain (all x values):

{-4, -2, 0, 1, 3}

Range (all y values):

{-2, 1, 2, -3, 1}

Ex. 2: Express the relation shown in the graph below as a set of ordered pairs. Then determine the domain and range. Then show the relation using a mapping.

Ordered Pairs:

(-4, -2), (-2, 1), (0, 2), (1, -3), and (3, 1)

Domain (all x values):

{-4, -2, 0, 1, 3}

Range (all y values):

{-2, 1, 2, -3}

-4

-2

0

1

3

-2

2

-3

1

In this relation, 3 maps to 1, 0 maps to 2, -2 maps to 1, -4 maps to -2 and 1 maps to -3.

Inverse of relation

• The inverse of any relation is obtained by switching the coordinates in each ordered pair of the relation. Thus the inverse of the relation {(2, 2), (-2, 3), (0, -1)} is the relation {(2, 2), (3, -2), (-1, 0)}. Notice that the domain of the relation becomes the range of the inverse and the range of the relation becomes the domain of the inverse.

Definition of the Inverse of a Relation

• Relation Q is the inverse of relation S if an only if for every ordered pair, (a, b) in S, there is an ordered pair (b, a) in Q.

Ex. 3: Express the relation shown in the mapping below as a set of ordered pairs. Write the inverse of this relation. Then determine the domain and range of the inverse.

Ordered Pairs:

(0, 4), (1, 5), (2, 6) and (3, 6)

Domain of inverse (all x values):

{4, 5, 6}

Range of inverse (all y values):

{0, 1, 2, 3}

0

1

2

3

4

5

6

In this relation, 3 maps to 1, 0 maps to 2, -2 maps to 1, -4 maps to -2 and 1 maps to -3.

Inverse of the relation:

(4, 0), (5, 1), (6, 2) and (6, 3)