1.1 Section Exercises

Verbal

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

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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?

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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?

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Algebraic

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

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

8. $5x+2y=10$

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

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

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

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

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

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

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

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

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

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

19. $2xy=1$

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

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

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

23. $x=±\sqrt{1-y}$

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24. $y=±\sqrt{1-x}$

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

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

For the following exercises, evaluate the function $\text{ }f\text{ }$ 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. $f\left(x\right)=2x-5$

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$\begin{array}{cccc}f\left(-3\right)=-11;& f\left(2\right)=-1;& f\left(-a\right)=-2a-5;& -f\left(a\right)=-2a+5;\text{ }\text{ }\text{ }\text{ }f\left(a+h\right)=2a+2h-5\end{array}$

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

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

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$\begin{array}{cccc}f\left(-3\right)=\sqrt{5}+5;& f\left(2\right)=5;& f\left(-a\right)=\sqrt{2+a}+5;& -f\left(a\right)=-\sqrt{2-a}-5;\text{ }\text{ }\text{ }\text{ }\text{ }f\left(a+h\right)=\end{array}$ $\sqrt{2-a-h}+5$

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

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

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$\begin{array}{cccc}f\left(-3\right)=2;& f\left(2\right)=1-3=-2;& f\left(-a\right)=|-a-1|-|-a+1|;& -f\left(a\right)=-|a-1|\text{ }+|a+1|;\text{ }\text{ }f\left(a+h\right)=\text{ }|a+h-1|-|a+h+1|\end{array}$

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

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

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$\frac{g\left(x\right)-g\left(a\right)}{x-a}=x+a+2,\text{ }x\ne a$

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

Evaluate $\text{ }k\left(2\right).$
Solve $\text{ }k\left(t\right)=7.$

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

Evaluate $\text{ }f\left(-2\right).$
Solve $\text{ }f\left(x\right)=-1.$

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a. $\text{ }f\left(-2\right)=14;\text{ }$ b. $\text{ }x=3$

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

Evaluate $\text{ }p\left(-3\right).$
Solve $\text{ }p\left(c\right)=2.$

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

Evaluate $\text{ }f\left(5\right).$
Solve $\text{ }f\left(x\right)=4.$

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a. $\text{ }f\left(5\right)=10;\text{ }$ b. $\text{ }x=-1\text{ }$ or $\text{ }x=4$

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

Evaluate $\text{ }f\left(7\right).$
Solve $\text{ }f\left(x\right)=4.$

39. Consider the relationship $\text{ }3r+2t=18.$

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

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a. $\text{ }f\left(t\right)=6-\frac{2}{3}t;\text{ }$ b. $\text{ }f\left(-3\right)=8;\text{ }$ c. $\text{ }t=6\text{ }$

Graphical

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

40.

41.

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43.

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45.

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52. Given the following graph,

Evaluate $\text{ }f\left(-1\right).$
Solve for $\text{ }f\left(x\right)=3.$

graph of a relation

53. Given the following graph,

Evaluate $\text{ }f\left(0\right).$
Solve for $\text{ }f\left(x\right)=-3.$

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a. $\text{ }f\left(0\right)=1;\text{ }$ b. $\text{ }f\left(x\right)=-3,\text{ }x=-2\text{ }$ or $\text{ }x=2\text{ }$

54. Given the following graph,

Evaluate $\text{ }f\left(4\right).$
Solve for $\text{ }f\left(x\right)=1.$

graph of a relation

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

55.

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Numeric

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

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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 $\text{ }y\text{ }$ as a function of $\text{ }x.$

63.

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

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64.

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

65.

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

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For the following exercises, use the function $\text{ }f\text{ }$ represented in (Figure).

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

66. Evaluate $\text{ }f\left(3\right).$

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

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$f\left(x\right)=1,\text{ }x=2$

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

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

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

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$\begin{array}{ccccc}f\left(-2\right)=14;& f\left(-1\right)=11;& f\left(0\right)=8;& f\left(1\right)=5;& f\left(2\right)=2\end{array}$

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

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

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$\begin{array}{ccccc}f\left(-2\right)=4;\text{ }& f\left(-1\right)=4.414;& f\left(0\right)=4.732;& f\left(1\right)=4.5;& f\left(2\right)=5.236\end{array}$

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

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

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$\begin{array}{ccccc}f\left(-2\right)=\frac{1}{9};& f\left(-1\right)=\frac{1}{3};& f\left(0\right)=1;& f\left(1\right)=3;& f\left(2\right)=9\end{array}$

For the following exercises, evaluate the expressions, given functions $f,\text{ }\text{ }g,$ and $\text{ }h:$

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

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

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

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Technology

For the following exercises, graph $\text{ }y={x}^{2}\text{ }$ 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. $\left[-10,\text{ 10}\right]$

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

graph of a parabola

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

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

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

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$\left[-0.001,\text{ 0}\text{.001}\right]$

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

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

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

For the following exercises, graph $\text{ }y=\sqrt{x}\text{ }$ 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. $\left[0,\text{ 100}\right]$

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

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

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

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

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

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

Real-World Applications

88. The amount of garbage, $\text{ }G,\text{ }$ produced by a city with population $\text{ }p\text{ }$ is given by $\text{ }G=f\left(p\right).\text{ }$ $G\text{ }$ is measured in tons per week, and $\text{ }p\text{ }$ 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 $\text{ }f.\text{ }$
Explain the meaning of the statement $\text{ }f\left(5\right)=2.$

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

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

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a. $\text{ }g\left(5000\right)=50;$ b. The number of cubic yards of dirt required for a garden of 100 square feet is 1.

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

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

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

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

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

Functions