Section Exercises

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. $\left\{\left(a,b\right),\text{ }\left(c,d\right),\text{ }\left(a,c\right)\right\}$

7. $\left\{\left(a,b\right),\left(b,c\right),\left(c,c\right)\right\}$

For the following exercises, determine whether the relation represents $y$ as a function of $x$.

8. $5x+2y=10$

9. $y={x}^{2}$

10. $x={y}^{2}$

11. $3{x}^{2}+y=14$

12. $2x+{y}^{2}=6$

13. $y=-2{x}^{2}+40x$

14. $y=\frac{1}{x}$

15. $x=\frac{3y+5}{7y - 1}$

16. $x=\sqrt{1-{y}^{2}}$

17. $y=\frac{3x+5}{7x - 1}$

18. ${x}^{2}+{y}^{2}=9$

19. $2xy=1$

20. $x={y}^{3}$

21. $y={x}^{3}$

22. $y=\sqrt{1-{x}^{2}}$

23. $x=\pm \sqrt{1-y}$

24. $y=\pm \sqrt{1-x}$

25. ${y}^{2}={x}^{2}$

26. ${y}^{3}={x}^{2}$

For the following exercises, evaluate the function $f$ at the indicated values $\text{ }f\left(-3\right),f\left(2\right),f\left(-a\right),-f\left(a\right),f\left(a+h\right)$.

27. $f\left(x\right)=2x - 5$

28. $f\left(x\right)=-5{x}^{2}+2x - 1$

29. $f\left(x\right)=\sqrt{2-x}+5$

30. $f\left(x\right)=\frac{6x - 1}{5x+2}$

31. $f\left(x\right)=|x - 1|-|x+1|$

32. Given the function $g\left(x\right)=5-{x}^{2}$, evaluate $\frac{g\left(x+h\right)-g\left(x\right)}{h},h\ne 0$.

33. Given the function $g\left(x\right)={x}^{2}+2x$, evaluate $\frac{g\left(x\right)-g\left(a\right)}{x-a},x\ne a$.

34. Given the function $k\left(t\right)=2t - 1:$

a. Evaluate $k\left(2\right)$.
b. Solve $k\left(t\right)=7$.

35. Given the function $f\left(x\right)=8 - 3x:$

a. Evaluate $f\left(-2\right)$.
b. Solve $f\left(x\right)=-1$.

36. Given the function $p\left(c\right)={c}^{2}+c:$

a. Evaluate $p\left(-3\right)$.
b. Solve $p\left(c\right)=2$.

37. Given the function $f\left(x\right)={x}^{2}-3x:$

a. Evaluate $f\left(5\right)$.
b. Solve $f\left(x\right)=4$.

38. Given the function $f\left(x\right)=\sqrt{x+2}:$

a. Evaluate $f\left(7\right)$.
b. Solve $f\left(x\right)=4$.

39. Consider the relationship $3r+2t=18$.

a. Write the relationship as a function $r=f\left(t\right)$.
b. Evaluate $f\left(-3\right)$.
c. Solve $f\left(t\right)=2$.

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

40.

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52. Given the following graph,

a. Evaluate $f\left(-1\right)$.
b. Solve for $f\left(x\right)=3$.

53. Given the following graph,

a. Evaluate $f\left(0\right)$.
b. Solve for $f\left(x\right)=-3$.

54. Given the following graph,

a. Evaluate $f\left(4\right)$.
b. Solve for $f\left(x\right)=1$.

For the following exercises, determine if the given graph is a one-to-one function.

55.

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59.

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

60. $\left\{\left(-1,-1\right),\left(-2,-2\right),\left(-3,-3\right)\right\}$

61. $\left\{\left(3,4\right),\left(4,5\right),\left(5,6\right)\right\}$

62. $\left\{\left(2,5\right),\left(7,11\right),\left(15,8\right),\left(7,9\right)\right\}$

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

63.

 $x$ 5 10 15 $y$ 3 8 14

64.

 $x$ 5 10 15 $y$ 3 8 8

65.

 $x$ 5 10 10 $y$ 3 8 14

For the following exercises, use the function $f$ represented in the table below.

 $x$ $f\left(x\right)$ 0 74 1 28 2 1 3 53 4 56 5 3 6 36 7 45 8 14 9 47

66. Evaluate $f\left(3\right)$.

67. Solve $f\left(x\right)=1$.

For the following exercises, evaluate the function $f$ at the values $f\left(-2\right),f\left(-1\right),f\left(0\right),f\left(1\right)$, and $f\left(2\right)$.

68. $f\left(x\right)=4 - 2x$

69. $f\left(x\right)=8 - 3x$

70. $f\left(x\right)=8{x}^{2}-7x+3$

71. $f\left(x\right)=3+\sqrt{x+3}$

72. $f\left(x\right)=\frac{x - 2}{x+3}$

73. $f\left(x\right)={3}^{x}$

For the following exercises, evaluate the expressions, given functions $f,g$, and $h:$

• $f\left(x\right)=3x - 2$
• $g\left(x\right)=5-{x}^{2}$
• $h\left(x\right)=-2{x}^{2}+3x - 1$

74. $3f\left(1\right)-4g\left(-2\right)$

75. $f\left(\frac{7}{3}\right)-h\left(-2\right)$

For the following exercises, graph $y={x}^{2}$ on the given viewing window. Determine the corresponding range for each viewing window. Show each graph.

76. $\left[-0.1,\text{ }0.1\right]$

77. $\left[-10,\text{ 10}\right]$

78. $\left[-100,100\right]$

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

79. $\left[-0.1,\text{ 0}\text{.1}\right]$

80. $\left[-10,\text{ 10}\right]$

81. $\left[-100,\text{ 100}\right]$

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

82. $\left[0,\text{ 0}\text{.01}\right]$

83. $\left[0,\text{ 100}\right]$

84. $\left[0,\text{ 10,000}\right]$

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

85. $\left[-0.001,\text{0.001}\right]$

86. $\left[-1000,\text{1000}\right]$

87. $\left[-1,000,000,\text{1,000,000}\right]$

88. The amount of garbage, $G$, produced by a city with population $p$ is given by $G=f\left(p\right)$. $G$ is measured in tons per week, and $p$ is measured in thousands of people.

a. 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 $f$.
b. Explain the meaning of the statement $f\left(5\right)=2$.

89. The number of cubic yards of dirt, $D$, needed to cover a garden with area $a$ square feet is given by $D=g\left(a\right)$.

a. A garden with area 5000 ft2 requires 50 yd3 of dirt. Express this information in terms of the function $g$.
b. Explain the meaning of the statement $g\left(100\right)=1$.

90. Let $f\left(t\right)$ be the number of ducks in a lake $t$ years after 1990. Explain the meaning of each statement:

a. $f\left(5\right)=30$
b. $f\left(10\right)=40$

91. Let $h\left(t\right)$ be the height above ground, in feet, of a rocket $t$ seconds after launching. Explain the meaning of each statement:

a. $h\left(1\right)=200$
b. $h\left(2\right)=350$

92. Show that the function $f\left(x\right)=3{\left(x - 5\right)}^{2}+7$ is not one-to-one.