9.2 Relations CORD Math Mrs. Spitz Fall 2006
Objectives Identify the domain, range and inverse of a relation, and Show relations as sets of ordered pairs and mappings
Assignment Pgs #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. xy x y
Ex. 1: Express the relation shown in the table below as a set of ordered pairs. Then determine the domain and range. xy 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} 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} 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)