1.9 Practice Questions

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?

Odd Number Verbal Solutions

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

6. [latex]\left\{\left(a,b\right),\text{ }\left(c,d\right),\text{ }\left(a,c\right)\right\}[/latex]
7. [latex]\left\{\left(a,b\right),\left(b,c\right),\left(c,c\right)\right\}[/latex]

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

8. [latex]5x+2y=10[/latex]
9. [latex]y={x}^{2}[/latex]
10. [latex]x={y}^{2}[/latex]
11. [latex]3{x}^{2}+y=14[/latex]
12. [latex]2x+{y}^{2}=6[/latex]
13. [latex]y=-2{x}^{2}+40x[/latex]
14. [latex]y=\frac{1}{x}[/latex]
15. [latex]x=\frac{3y+5}{7y-1}[/latex]
16. [latex]x=\sqrt{1-{y}^{2}}[/latex]
17. [latex]y=\frac{3x+5}{7x-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=±\sqrt{1-y}[/latex]
24. [latex]y=±\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]\text{}f\text{}[/latex] at the indicated values [latex]\text{ }f\left(-3\right),f\left(2\right),f\left(-a\right),-f\left(a\right),f\left(a+h\right)[/latex].

27. [latex]f\left(x\right)=2x-5[/latex]
28. [latex]f\left(x\right)=-5{x}^{2}+2x-1[/latex]
29. [latex]f\left(x\right)=\sqrt{2-x}+5[/latex]
30. [latex]f\left(x\right)=\frac{6x-1}{5x+2}[/latex]
31. [latex]f\left(x\right)=|x-1|-|x+1|[/latex]
32. Given the function [latex]\text{}g\left(x\right)=5-{x}^{2},\text{}[/latex] evaluate [latex]\text{}\frac{g\left(x+h\right)-g\left(x\right)}{h},\text{}h\ne 0[/latex]
33. Given the function [latex]\text{}g\left(x\right)={x}^{2}+2x,\text{}[/latex] evaluate [latex]\text{}\frac{g\left(x\right)-g\left(a\right)}{x-a},\text{}x\ne a[/latex]
34. Given the function [latex]\text{}k\left(t\right)=2t-1:[/latex]
    1. Evaluate [latex]\text{}k\left(2\right)[/latex]
    2. Solve [latex]\text{}k\left(t\right)=7[/latex]
35. Given the function [latex]\text{}f\left(x\right)=8-3x:[/latex]
    1. Evaluate [latex]\text{}f\left(-2\right)[/latex]
    2. Solve [latex]\text{}f\left(x\right)=-1[/latex]
36. Given the function [latex]\text{}p\left(c\right)={c}^{2}+c:[/latex]
    1. Evaluate [latex]\text{}p\left(-3\right)[/latex]
    2. Solve [latex]\text{}p\left(c\right)=2[/latex]
37. Given the function [latex]\text{}f\left(x\right)={x}^{2}-3x:[/latex]
    1. Evaluate [latex]\text{}f\left(5\right)[/latex]
    2. Solve [latex]\text{}f\left(x\right)=4[/latex]
38. Given the function [latex]\text{}f\left(x\right)=\sqrt{x+2}:[/latex]
    1. Evaluate [latex]\text{}f\left(7\right)[/latex]
    2. Solve [latex]\text{}f\left(x\right)=4[/latex]
39. Consider the relationship [latex]\text{}3r+2t=18[/latex]
    1. Write the relationship as a function [latex]\text{}r=f\left(t\right)[/latex]
    2. Evaluate [latex]\text{}f\left(-3\right)[/latex]
    3. Solve [latex]\text{}f\left(t\right)=2[/latex]

Odd Number Algebraic Solutions

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

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51. 
52. Given the following graph,
    1. Evaluate [latex]\text{}f\left(-1\right)[/latex].
    2. Solve for [latex]\text{}f\left(x\right)=3[/latex].
       
53. Given the following graph,
    1. Evaluate [latex]\text{}f\left(0\right)[/latex].
    2. Solve for [latex]\text{}f\left(x\right)=-3[/latex].
       
54. Given the following graph,
    1. Evaluate [latex]\text{}f\left(4\right)[/latex].
    2. Solve for [latex]\text{}f\left(x\right)=1[/latex].
       

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

Odd Number Graphical Solutions

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

60. [latex]\left\{\left(-1,-1\right),\left(-2,-2\right),\left(-3,-3\right)\right\}[/latex]
61. [latex]\left\{\left(3,4\right),\left(4,5\right),\left(5,6\right)\right\}[/latex]
62. [latex]\left\{\left(2,5\right),\left(7,11\right),\left(15,8\right),\left(7,9\right)\right\}[/latex]

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

63. [latex]x[/latex]	5	10	15
    [latex]y[/latex]	3	8	14

64. [latex]x[/latex]	5	10	15
    [latex]y[/latex]	3	8	8

65. [latex]x[/latex]	5	10	10
    [latex]y[/latex]	3	8	14

For the following exercises, use the function [latex]\text{}f\text{}[/latex] represented in Table 1-14.

66. Evaluate [latex]\text{}f\left(3\right)[/latex].
67. Solve [latex]\text{}f\left(x\right)=1[/latex].

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

68. [latex]f\left(x\right)=4-2x[/latex]
69. [latex]f\left(x\right)=8-3x[/latex]
70. [latex]f\left(x\right)=8{x}^{2}-7x+3[/latex]
71. [latex]f\left(x\right)=3+\sqrt{x+3}[/latex]
72. [latex]f\left(x\right)=\frac{x-2}{x+3}[/latex]
73. [latex]f\left(x\right)={3}^{x}[/latex]

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

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

74. [latex]3f\left(1\right)-4g\left(-2\right)[/latex]
75. [latex]f\left(\frac{7}{3}\right)-h\left(-2\right)[/latex]

Odd Number Numeric Solutions

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

76. [latex]\left[-0.1,\text{ }0.1\right][/latex]
77. [latex]\left[-10,\text{ 10}\right][/latex]
78. [latex]\left[-100,100\right][/latex]

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

79. [latex]\left[-0.1,\text{ 0}\text{.1}\right][/latex]
80. [latex]\left[-10,\text{ 10}\right][/latex]
81. [latex]\left[-100,\text{ 100}\right][/latex]

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

82. [latex]\left[0,\text{ 0}\text{.01}\right][/latex]
83. [latex]\left[0,\text{ 100}\right][/latex]
84. [latex]\left[0,\text{ 10,000}\right][/latex]

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

85. [latex]\left[-0.001,\text{0.001}\right][/latex]
86. [latex]\left[-1000,\text{1000}\right][/latex]
87. [latex]\left[-1,000,000,\text{1,000,000}\right][/latex]

Odd Number Technology Solutions

88. The amount of garbage, [latex]\text{}G,\text{}[/latex] produced by a city with population [latex]\text{}p\text{}[/latex] is given by [latex]\text{}G=f\left(p\right)\text{}[/latex]. [latex]G\text{}[/latex] is measured in tons per week, and [latex]\text{}p\text{}[/latex] is measured in thousands of people.
    1. 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]\text{}f\text{}[/latex].
    2. Explain the meaning of the statement [latex]\text{}f\left(5\right)=2[/latex].
89. The number of cubic yards of dirt, [latex]\text{}D,\text{}[/latex] needed to cover a garden with area [latex]\text{}a\text{}[/latex] square feet is given by [latex]\text{}D=g\left(a\right)[/latex].
    1. A garden with area 5000 ft² requires 50 yd³ of dirt. Express this information in terms of the function [latex]\text{}g[/latex].
    2. Explain the meaning of the statement [latex]\text{}g\left(100\right)=1[/latex].
90. Let [latex]\text{}f\left(t\right)\text{}[/latex] be the number of ducks in a lake [latex]\text{}t\text{}[/latex] years after 1990. Explain the meaning of each statement:
    1. [latex]f\left(5\right)=30[/latex]
    2. [latex]f\left(10\right)=40[/latex]
91. Let [latex]\text{}h\left(t\right)\text{}[/latex] be the height above ground, in feet, of a rocket [latex]\text{}t\text{}[/latex] seconds after launching. Explain the meaning of each statement:
    1. [latex]h\left(1\right)=200[/latex]
    2. [latex]h\left(2\right)=350[/latex]
92. Show that the function [latex]\text{}f\left(x\right)=3{\left(x-5\right)}^{2}+7\text{}[/latex] is not one-to-one.

Odd Number Real-World Application Solutions