74 CHAPTER 2 Functions and Their Graphs 15. The expression ( ) ( ) + − f x h f x h is called the of f. 16. When written as ( ) = y f x , a function is said to be defined . In Problems 51–70, find the domain of each function. 51. ( ) = − + f x x5 4 52. ( ) = + f x x 2 2 53. ( ) = + + f x x x 1 2 8 2 54. ( ) = + f x x x 1 2 2 55. ( ) = − g x x x 16 2 56. ( ) = − h x x x 2 4 2 57. ( ) = − + F x x x x 2 3 58. ( ) = + − G x x x x 4 4 3 Skill Building In Problems 17 and 18, a relation expressed verbally is given. (a) What is the domain and the range of the relation? (b) Express the relation using a mapping. (c) Express the relation as a set of ordered pairs. 17. The density of a gas under constant pressure depends on temperature. Holding pressure constant at 14.5 pounds per square inch, a chemist measures the density of an oxygen sample at temperatures of 0, 22, 40, 70, and ° 100 C and obtains densities of 1.411, 1.305, 1.229, 1.121, and 1.031 kg m , 3 respectively. 18. A researcher wants to investigate how weight depends on height among men in Europe. She visits five regions in Europe and determines the average heights in those regions to be 1.80, 1.78, 1.77, 1.77, and 1.80 meters. The corresponding average weights are 87.1, 86.9, 83.0, 84.1, and 86.4 kg, respectively. 19. Elvis Person Colleen Kaleigh Marissa Jan. 8 Birthday Mar. 15 Sept. 17 In Problems 19–30, find the domain and range of each relation. Then determine whether the relation represents a function. 20. Bob Father Darius Chuck Beth Daughter Diane Imani Marcia 22. 9th-12th grade High School Graduate Some College Bachelor's Degree Level of Education $682 $853 $935 $1432 Weekly Average Income 21. 20 Hours Hours Worked 30 Hours 40 Hours $300 Salary $350 $540 $880 23. ( ) ( ) ( ) ( ) { } − 2, 6 , 3, 6 , 4, 9 , 2, 10 24. ( ) ( ) ( ) ( ) { } − − 2, 5 , 1, 3 , 3, 7 , 4, 12 25. {( )( )( )( )} 1, 3 , 2, 3 , 3, 3 , 4, 3 26. ( ) ( ) ( ) ( ) { } − 0, 2, 1,3, 2,3, 3,7 27. ( ) ( ) ( ) ( ) { } − 3, 3 , 3, 5 , 0, 1 , 4, 6 28. ( ) ( ) ( ) ( ) ( ) { } − − − − − 4, 4 , 3, 3 , 2, 2 , 1, 1 , 4, 0 29. ( ) ( ) ( ) ( ) { } − − 1,8, 0,3, 2, 1, 4,3 30. ( ) ( ) ( ) ( ) { } − − 2, 16 , 1, 4 , 0, 3 , 1, 4 In Problems 31–42, determine whether the equation defines y as a function of x. 31. = − + y x x 2 3 4 2 32. = y x3 33. = y x 1 34. = y x 35. = − x y 8 2 2 36. = ± − y x 1 2 37. = x y2 38. + = x y 1 2 39. = y x 3 40. = − + y x x 3 1 2 41. = + y x2 3 42. − = x y4 1 2 2 In Problems 43–50, find the following for each function: (a) ( ) f 0 (b) ( ) f 1 (c) ( ) − f 1 (d) ( ) − f x (e) ( ) −f x (f) ( ) + f x 1 (g) ( ) f x2 (h) ( ) + f x h 43. ( ) = + − f x x x 3 2 4 2 44. ( ) = − + − f x x x 2 1 2 45. ( ) = + f x x x 1 2 46. ( ) = − + f x x x 1 4 2 47. ( ) = + f x x 4 48. ( ) = + f x x x 2 49. ( ) = + − f x x x 2 1 3 5 50. ( ) ( ) = − + f x x 1 1 2 2
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