2. © William James Calhoun, 2001 5-2: Relations OBJECTIVES: You will be able to identify the domain, range, and inverse of a relation, and show relations as sets of ordered pairs, tables, mappings, and graphs. In the United States, the manatee, an aquatic mammal, is considered to be endangered. In the table below, you will find the number of manatees that have been found dead since 1981.

3. © William James Calhoun, 2001 The manatee data could also be represented by a set of ordered pairs, as shown in the list below. Each first coordinate is the year, and the second coordinate is the number of manatees. {(1981, 116), (1982, 114), (1983, 81), (1984, 128), (1985, 119), (1986, 122), (1987, 114), (1988, 133), (1989, 168), (1990, 206), (1991, 174), (1992, 163), (1993, 145), (1994, 193), (1995, 201), (1996, 415), (1997, 242), (1998, 231)} Each ordered pair can then be graphed. 5-2: Relations

4. © William James Calhoun, 2001 5-2: Relations relation - set of ordered pairs (like the ones from the example) New terms: domain - the set of first coordinates of the ordered pairs This is also considered the independent variable. It changes due to human input or time. In math, it is the x-value. range - the set of second coordinates of the ordered pairs This is also considered the dependent variable. It changes because the domain (independent variable) changed. In math, it is the y-value. Remember the term:

5. © William James Calhoun, 2001 The domain of a relation is the set of all first coordinates from the ordered pairs in the relation. The range of the relation is the set of all second coordinates from the ordered pairs. 5.2.1 DEFINITION OF THE DOMAIN AND RANGE OF A RELATION 5-2: Relations Remember, the domain is the x-value - the independent variable. The range is the y-value - the dependent variable.

6. © William James Calhoun, 2001 We can identify the domain and range from the manatee data. {(1981, 116), (1982, 114), (1983, 81), (1984, 128), (1985, 119), (1986, 122), (1987, 114), (1988, 133), (1989, 168), (1990, 206), (1991, 174), (1992, 163), (1993, 145), (1994, 193), (1995, 201), (1996, 415), (1997, 242), (1998, 231)} All of the dates (the first values) are in the domain. {(1981, 116), (1982, 114), (1983, 81), (1984, 128), (1985, 119), (1986, 122), (1987, 114), (1988, 133), (1989, 168), (1990, 206), (1991, 174), (1992, 163), (1993, 145), (1994, 193), (1995, 201), (1996, 415), (1997, 242), (1998, 231)} The number of manatee (the second values) are in the range. {(1981, 116), (1982, 114), (1983, 81), (1984, 128), (1985, 119), (1986, 122), (1987, 114), (1988, 133), (1989, 168), (1990, 206), (1991, 174), (1992, 163), (1993, 145), (1994, 193), (1995, 201), (1996, 415), (1997, 242), (1998, 231)} The domain and range can be listed: D: {1981, 1982, 1983, 1984, 1985, 1986, 1987, 1988, 1989, 1990, 1991, 1992, 1993, 1994, 1995, 1996, 1997, 1998} R: {116, 81, 128, 119, 122, 133, 168, 206, 174, 163, 145, 193, 201, 415, 242, 231} There are many ways to represent data. We looked at the manatee data as ordered pairs, a table, and a graph. Data can also be represented in a mapping. 5-2: Relations

7. © William James Calhoun, 2001 5-2: Relations Term: Here is an example of the different ways to represent a relation. The relation used is: {(3, 3), (-1, 4), (0, -4)} Ordered PairsTableGraphMapping (3, 3) (-1, 4) (0, -4) mapping - an illustration how each element of the domain is paired with an element in the range.

8. © William James Calhoun, 2001 5-2: Relations EXAMPLE 1: Represent the relation shown in the graph at the right as: A. a set of ordered pairs, B. a table, and C. a mapping. D. The determine the domain and range. A. Determine the coordinates of each point, and write the ordered pairs in relation brackets. It helps to start on the left and work your way right. {(-3, 3),(-1, 2),(1, 1),(1, 3),(3, -2),(4, -2)} B. Make an x-y table and put the values in: C. Write the x’s - no repeats. -3 1 3 4 Write the y’s - no repeats. 3 2 1 -2 Draw arrows to match pairs. D. D = {-3, -1, 1, 3, 4} R = {-2, 1, 2, 3} NOTICE: The domain and range have no repeats and are in order from least to greatest. Draw ovals around sets.

9. © William James Calhoun, 2001 You can find the inverse of a relation by simply switching the domain and range for all ordered pairs. Relation (1, 4) (-3, 2) (7, -9) Inverse (4, 1) (2, -3) (-9, 7) 5-2: Relations Notice the inverse is the mirror image of an up-to-the-left diagonal line through the origin.

10. © William James Calhoun, 2001 5-2: Relations The mapping shows the relation {(1, -3), (2, -2), (3, -1), (4, -1)}. To get the inverse, switch the x- and y-values: {(-3, 1), (-2, 2), (-1, 3), (-1, 4)}. Then map the inverse. EXAMPLE 2: Express the relation shown in the mapping as a set of ordered pairs. Write the inverse of the relation and draw a mapping to model the inverse. X 1 2 3 4 Y -3 -2 -3 -2 12341234 Relation Q is the inverse of relation S if and only if for every ordered pair (a, b) in S, there is an ordered pair (b, a) in Q. 5.2.2 DEFINITION OF THE INVERSE OF A RELATION

11. © William James Calhoun, 2001 5-2: Relations EXAMPLE 3: Express the relation shown in the mapping as a set of ordered pairs. Identify the domain and range. Then write the inverse of this relation. XY 3 4 -6 -4 2 Relation = {(3, 2), (4, -6), (3, -4), (-1, -6)} Domain = {-1, 3, 4} Range = {-6, -4, 2} Inverse = {(2, 3), (-6, 4), (-4, 3), (-6, -1)} BIG HINT: Your 3-Strikes and Homework should look like this answer!!!

12. © William James Calhoun, 2001 5-2: Relations HOMEWORK Page 267 #17 - 37 odd