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5-2: Relations

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.

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5-2: Relations

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  1. 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.

  2. 5-2: Relations 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.

  3. 5-2: Relations Remember the term: • 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.

  4. 5.2.1 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 in the relation. The range of the relation is the set of all second coordinates from the ordered pairs. 5-2: Relations Remember, the domain is the x-value - the independent variable. The range is the y-value - the dependent variable.

  5. 5-2: Relations We can identify the domain and range from the manatee data. 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)} {(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)} {(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. 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.

  6. 5-2: Relations Term: • mapping - an illustration how each element of the domain is paired with an element in the range. Here is an example of the different ways to represent a relation. The relation used is: {(3, 3), (-1, 4), (0, -4)} Ordered Pairs Table Graph Mapping (3, 3) (-1, 4) (0, -4)

  7. 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 1 3 4 Write the y’s - no repeats. 3 2 1 -2 Draw ovals around sets. 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.

  8. 5-2: Relations 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) Notice the inverse is the mirror image of an up-to-the-left diagonal line through the origin.

  9. 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 -1 1 2 3 4 -3 -2 -1 5.2.2 DEFINITION OF THE INVERSE OF A RELATION 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: 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.

  10. 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. X Y -1 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!!!

  11. 5-2: Relations HOMEWORK Page 267 #17 - 37 odd

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