1. Relations Equations as Relations

2. Warm-up Problems (4)

4. Warm-up ANSWERS

5. Warm-up Problems (4)

7. Vocabulary Relation – A set of ordered pairs Domain – The set of all x coordinates in a relation Range – The set of all y coordinates in a relation Mapping – Illustrates how each domain element is paired with a range element

8. Ways to show a relation

9. Example 3-1a Express the relation {(4, 3), (–2, –1), (–3, 2), (2, –4), (0, –4)} as a table, a graph and a mapping. Table List the set of x -coordinates in the first column and the corresponding y -coordinates in the second column. –40 2 2–3 –1–2 34 yx

10. Example 3-1a Graph Graph each ordered pair on a coordinate plane.

11. Example 3-1b Mapping List the x values in set X and the y values in set Y. Draw an arrow from each x value in X to the corresponding y value in Y.

12. Example 3-1b Determine the domain and range. Answer:The domain for this relation is {–3, –2, 0, 2, 4}. The range is {–4, –1, 2, 3}.

13. Example 3-1c a.Express the relation {(3, –2), (4, 6), (5, 2), (–1, 3)} as a table, a graph, and a mapping. b.Determine the domain and range. Answer: xy 3–2 46 52 –13 Answer: D = {–1, 3, 4, 5} ; R = {–2, 2, 3, 6}

14. Example 3-2a Opinion Polls The table shows the percent of people satisfied with the way things were going in the U.S. at the time of the survey. Year 1992199519982001 Percent Satisfied 21326051 Determine the domain and range of the relation. Answer: The domain is {1992, 1995, 1998, 2001}. The range is {21, 32, 51, 60}.

15. Example 3-2b Graph the data. The values of the x-axis need to go from 1992 to 2001. It is not practical to begin the scale at 0. Begin at 1992 and extend to 2001 to include all of the data. The units can be 1 unit per grid square. The values on the y-axis need to go from 21 to 60. In this case it is possible to begin the scale at 0. Begin at 0 and extend to 70. You can use units of 10.

16. Example 3-2c What conclusions might you make from the graph of the data? Answer:Americans became more satisfied with the country from 1992-1998, but the percentage dropped from 1998-2001.

17. Example 3-2d Endangered Species The table shows the approximate world population of the Indian Rhinoceros from 1982 to 1998. Indian Rhinoceros Population Year 19821986199019941998 Population 10001700 19002100 a. Determine the domain and range of the relation. Answer: D = {1982, 1986, 1990, 1994, 1998} R = {1000, 1700, 1900, 2100}

18. Example 3-2d b. Graph the data. Answer:

19. Example 3-2d c. What conclusions might you make from the graph of the data? Answer: The population of the Indian rhinoceros has been increasing since 1982.

20. Vocabulary Inverse – Obtained from a relation by switching the coordinates of each ordered pair

21. Example 3-3a Express the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation. Answer: {(5, 1), (7, 2), (4, –9), (0, 2)} Relation Notice that both 7 and 0 in the domain are paired with 2 in the range. Inverse Exchange X and Y in each ordered pair to write the inverse relation. Answer: {(1, 5), (2, 7), (–9, 4), (2, 0)}

22. Example 3-3b Express the relation shown in the mapping as a set of ordered pairs. Then write the inverse of the relation. Answer:Relation: {(3, 2), (–4, 1), (5, 2)} Inverse: {(2, 3), (1, –4), (2, 5)}

23. Homework Quiz

26. Equations as Relations

27. Graphing Range values can have more than one arrow pointing to them. Can Domain values have more then one arrow pointing from them? No

28. Graphing The solution set of an equation are the ordered pairs that result in a true statement when substituted into the equation.

29. Example 4-1a Find the solution set for given the replacement set {(–5, 0), (–3, –2), (2, 13), (4, 19)}. Make a table. Substitute each ordered pair into the equation. 0–5 true false 194 132 –2–3 True or False? yx

30. Example 4-1a The ordered pairs ( –3, –2 ), ( 2, 13 ) and ( 4, 19 ) result in true statements. Answer:The solution set is {(–3, –2), (2, 13), (4, 19)}.

31. Example 4-1b Find the solution set for given the replacement set {(3, 1), (6, 8), (1, 5), (–1, 1)}. Answer: {(1, 5), (–1, –1)}

32. Example 4-2a Solve if the domain is {–2, 0, 3, 5, 8}. Make a table. The values of c come from the domain. Substitute each value of c into the equation to determine the values of d in the range. c8 – c8 – cd (c, d)(c, d) –28 – (–2)10 (–2, 10) 08 – 08 – 08 (0, 8) 38 – 38 – 35 (3, 5) 58 – 58 – 53 (5, 3) 88 – 88 – 80 (8, 0) Answer: The solution set is {(–2, 10), (0, 8), (3, 5), (5, 3), (8, 0)}.

33. Example 4-2b Answer: {(–1, –6), (0, –4), (2, 0), (5, 6)} Solve if the domain is {–1, 0, 2, 5}.

34. Example 4-3a Solve if the domain is {0, 1, 2, 3}. Graph the solution set. First solve the equation for y in terms of x. This makes creating a table of values easier. Original equation Subtract 9x from each side. Simplify. Divide each side by 3. Simplify.

35. Example 4-3a Substitute each value of x from the domain to determine the corresponding values of y in the range. x5 – 3x5 – 3xy (x, y)(x, y) 05 – 3(0)5 (0, 5) 15 – 3(1)2 (1, 2) 25 – 3(2)–1 (2, –1) 35 – 3(3)–4 (3, –4) Answer: The solution set is {(0, 5), (1, 2), (2, –1), (3, –4)}. Graph the ordered pairs.

36. Example 4-3b Solve if the domain is {0, 1, 2, 3}. Graph the solution set. Answer: {(0, 4), (1, 1), (2, –2), (3, –5)}

37. Example 4-4a Travel In 2002, 12 countries in Europe made the switch to a single currency, the euro. Suppose the exchange rate between U.S. dollars and euros is one dollar = 1.11 euros. The equationcan be used to convert U.S. dollars to euros. If a traveler is going to spend the following amounts per day while in Europe, find the equivalent U.S. dollars for the amounts given in euros (EUR) Graph the ordered pairs. Hotel90 EUR Food50 EUR Transportation30 EUR Gifts20 EUR

38. PlanThe values of E, {90, 50, 30, 20}, are the domain. Use the equation to find the values for range. Example 4-4a ExploreIn the equation, D represents U.S. dollars and E represents the euro. However, we are given values in euros and want to find values in dollars. Solve the equation for D since the values for D depend on the given values of E. Original equation Divide each side by 1.11. Simplify and round to the nearest hundredth.

39. Example 4-4a SolveMake a table of values. Substitute each value of E from the domain to determine the corresponding values of D. Round to the nearest dollar. E0.90ED (E, D)(E, D) 900.90(90)$81 (90, 81) 500.90(50)$45 (50, 45) 300.90(30)$27 (30, 27) 200.90(20)$18 (20, 18)

40. Example 4-4a Graph the ordered pairs. Answer: Notice that the values for the independent variable E are graphed along the horizontal axis, and the values for the dependent variable D are graphed along the vertical axis.

41. Example 4-4a The equivalent amounts in U.S. dollars for the given amounts in euros is shown in the table. ExamineLook at the values in the range. The cost in dollars is lower than the cost in euros. Do the results make sense? ExpenseEurosDollars Hotel 9081 Food 5045 Transportation 3027 Gifts 2018

42. Example 4-4b Travel The exchange rate between U.S. dollars and Swiss francs is one dollar = 1.68 Swiss francs. The equationcan be used to convert U.S. dollars to Swiss francs F. If a traveler is going to spend the following amounts per day while he is in Switzerland, find the equivalent U.S. dollars for the amounts given. Graph the ordered pairs. Hotel 80 francs $48 Food 45 francs $27 Transportation 35 francs $21 Gifts 15 francs $9 Answer:

43. Example 4-4b

44. Practice

45. Homework 4-3 Relations 4-4 Equations as Relations All Problems