1. What is the difference between a relation and a function?

2. What is the difference between the input and the output of a function?

3. Why does the vertical line test tell us whether the graph of a relation represents a function?

4. How can you determine if a relation is a one-to-one function?

5. Why does the horizontal line test tell us whether the graph of a function is one-to-one?

For the following exercises, determine whether the relation represents a function.

6. [latex]{(a,b), (c,d), (a,c)}[/latex]

7. [latex]{(a,b),(b,c),(c,c)}[/latex]

For the following exercises, determine whether the relation represents [latex]y[/latex] as a function of [latex]x[/latex].

8. [latex]5x+2y=10[/latex]

9. [latex]y=x^{2}[/latex]

10. [latex]x=y^{2}[/latex]

11. [latex]3x^{2}+y=14[/latex]

12. [latex]2x+y^{2}=6[/latex]

13. [latex]y=−2x^{2}+40x[/latex]

14. [latex]y=1x[/latex]

15. [latex]x=3y+57y−1[/latex]

16. [latex]x=\sqrt{1−y^2}[/latex]

17. [latex]y=3x+57x−1[/latex]

18. [latex]x^2+y^2=9[/latex]

19. [latex]2xy=1[/latex]

20. [latex]x=y^3[/latex]

21. [latex]y=x^3[/latex]

22. [latex]y=\sqrt{1−x^2}[/latex]

23. [latex]x=\pm\sqrt{1−y}[/latex]

24. [latex]y=\pm\sqrt{1−x}[/latex]

25. [latex]y^2=x^2[/latex]

26. [latex]y^3=x^2[/latex]

For the following exercises, evaluate the function [latex]f[/latex] at the indicated values [latex]f(−3),f(2),f(−a),−f(a),f(a+h)[/latex].

27. [latex]f(x)=2x−5[/latex]

28. [latex]f(x)=−5x^2+2x−1[/latex]

29. [latex]f(x)=\sqrt{2−x}+5[/latex]

30. [latex]f(x)=6x−15x+2[/latex]

31. [latex]f(x)=|x−1|−|x+1|[/latex]

32. Given the function [latex]g(x)=5−x^{2}[/latex],evaluate [latex]g(x+h)−g(x)h,h\ne{0}[/latex].

33. Given the function [latex]g(x)=x^{2}+2x[/latex],evaluate [latex]g(x)−g(a)x−a,x\ne{a}[/latex].

34. Given the function [latex]k(t)=2t−1[/latex]:

Evaluate [latex]k(2)[/latex].

Solve [latex]k(t)=7[/latex].

35. Given the function [latex]f(x)=8−3x[/latex]:
Evaluate [latex]f(−2)[/latex].
Solve [latex]f(x)=−1[/latex].

36. Given the function [latex]p(c)=c^2+c[/latex]:
Evaluate [latex]p(−3)[/latex].
Solve [latex]p(c)=2[/latex].

37. Given the function [latex]f(x)=x^2−3x[/latex]:
Evaluate [latex]f(5)[/latex].
Solve [latex]f(x)=4[/latex].

38. Given the function [latex]f(x)=/sqrt{x+2}[/latex]:
Evaluate [latex]f(7)[/latex].
Solve [latex]f(x)=4[/latex].

39. Consider the relationship [latex]3r+2t=18[/latex].
Write the relationship as a function [latex]r=f(t)[/latex].
Evaluate [latex]f(−3)[/latex].
Solve [latex]f(t)=2[/latex].

For the following exercises, use the vertical line test to determine which graphs show relations that are functions.

52. Given the following graph,
Evaluate [latex]f(−1)[/latex].
Solve for [latex]f(x)=3[/latex].
Graph of relation.
53. Given the following graph,
Evaluate [latex]f(0[/latex]).
Solve for [latex]f(x)=−3[/latex]. Graph of relation.
54. Given the following graph,
Evaluate [latex]f(4)[/latex].
Solve for [latex]f(x)=1[/latex].
Graph of relation.
For the following exercises, determine if the given graph is a one-to-one function.

55. Graph of a circle.
56. Graph of a parabola.
57. Graph of a rotated cubic function.
58. Graph of half of 1/x.
59. Graph of a one-to-one function.

For the following exercises, determine whether the relation represents a function.

60. [latex]{(−1,−1),(−2,−2),(−3,−3)}[/latex]

61. [latex]{(3,4),(4,5),(5,6)}[/latex]

62. [latex]{(2,5),(7,11),(15,8),(7,9)}[/latex]

For the following exercises, determine if the relation represented in table form represents [latex]y[/latex] as a function of [latex]x[/latex].

63.
x 5 10 15
y 3 8 14

64.
x 5 10 15
y 3 8 8

65.
For the following exercises, use the function [latex]f[/latex] represented in the table below.

[latex]x[/latex]	[latex]f(x)[/latex]
0	74
1	28
2	1
3	53
4	56
5	3
6	36
7	45
8	14
9	47

66. Evaluate [latex]f(3)[/latex].

67. Solve [latex]f(x)=1[/latex].

For the following exercises, evaluate the function [latex]f[/latex] at the values [latex]f(−2),f(−1),f(0),f(1),\text{ and }f(2)[/latex].

68. [latex]f(x)=4−2x[/latex]

69. [latex]f(x)=8−3x[/latex]

70. [latex]f(x)=8x^2−7x+3[/latex]

71. [latex]f(x)=3+\sqrt{x+3}[/latex]

72. [latex]f(x)=x−2x+3[/latex]

73. [latex]f(x)=3x[/latex]

For the following exercises, evaluate the expressions, given functions [latex]f,g,\text{ and }h[/latex]:

[latex]f(x)=3x−2[/latex]
[latex]g(x)=5−x^2[/latex]
[latex]h(x)=−2x^2+3x−1[/latex]

74. [latex]3f(1)−4g(−2)[/latex]

75. [latex]f(73)−h(−2)[/latex]

For the following exercises, graph [latex]y=x^2[/latex] on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

76. [latex][−0.1, 0.1][/latex]

77. [latex][−10, 10][/latex]

78. [latex][−100,100][/latex]

For the following exercises, graph [latex]y=x^{3}[/latex] on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

79. [latex][−0.1, 0.1][/latex]

80. [latex][−10, 10][/latex]

81. [latex][−100, 100][/latex]

For the following exercises, graph [latex]y=\sqrt{x}[/latex] on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

82. [latex][0, 0.01][/latex]
83. [latex][0, 100][/latex]
84. [latex][0, 10,000][/latex]

For the following exercises, graph [latex]y=x\sqrt{3}[/latex] on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

85. [latex][−0.001,0.001][/latex]
86. [latex][−1000,1000][/latex]
87. [latex][−1,000,000,1,000,000][/latex]

88. The amount of garbage, [latex]G[/latex], produced by a city with population [/latex]p[/latex] is given by [/latex]G=f(p)[/latex]. [latex]G[/latex] is measured in tons per week, and [latex]p[/latex] is measured in thousands of people.
The town of Tola has a population of 40,000 and produces 13 tons of garbage each week. Express this information in terms of the function [latex]f[/latex].
Explain the meaning of the statement [latex]f(5)=2[/latex].

89. The number of cubic yards of dirt, [latex]D[/latex],needed to cover a garden with area a square feet is given by [latex]D=g(a)[/latex].
A garden with area 5000 ft² requires 50 yd³ of dirt. Express this information in terms of the function [latex]g[/latex].
Explain the meaning of the statement [latex]g(100)=1[/latex].

90. Let [latex]f(t)[/latex] be the number of ducks in a lake [latex]t[/latex] years after 1990. Explain the meaning of each statement:
[latex]f(5)=30[/latex]
[latex]f(10)=40[/latex]

91. Let [latex]h(t)[/latex] be the height above ground, in feet, of a rocket [latex]t[/latex] seconds after launching. Explain the meaning of each statement:
[latex]h(1)=200[/latex]
[latex]h(2)=350[/latex]

92. Show that the function [latex]f(x)=3(x−5)^2+7[/latex] is not one-to-one.