1. 9.2 Relations CORD Math Mrs. Spitz Fall 2006

2. Objectives Identify the domain, range and inverse of a relation, and Show relations as sets of ordered pairs and mappings

3. Assignment Pgs 361-363 #4-41

4. 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.

5. 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. xy 22 -23 0 2 -2 0 2 3 x y

6. Ex. 1: Express the relation shown in the table below as a set of ordered pairs. Then determine the domain and range. xy 05 23 1-4 -33 -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}

7. 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}

8. 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.

9. 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.

10. 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.

11. 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} 01230123 456456 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)