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]\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]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]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=\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]\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]g\left(x\right)=5-{x}^{2}[/latex], evaluate [latex]\frac{g\left(x+h\right)-g\left(x\right)}{h},h\ne 0[/latex].
33. Given the function [latex]g\left(x\right)={x}^{2}+2x[/latex], evaluate [latex]\frac{g\left(x\right)-g\left(a\right)}{x-a},x\ne a[/latex].
34. Given the function [latex]k\left(t\right)=2t - 1:[/latex]
a. Evaluate [latex]k\left(2\right)[/latex].
b. Solve [latex]k\left(t\right)=7[/latex].
35. Given the function [latex]f\left(x\right)=8 - 3x:[/latex]
a. Evaluate [latex]f\left(-2\right)[/latex].
b. Solve [latex]f\left(x\right)=-1[/latex].
36. Given the function [latex]p\left(c\right)={c}^{2}+c:[/latex]
a. Evaluate [latex]p\left(-3\right)[/latex].
b. Solve [latex]p\left(c\right)=2[/latex].
37. Given the function [latex]f\left(x\right)={x}^{2}-3x:[/latex]
a. Evaluate [latex]f\left(5\right)[/latex].
b. Solve [latex]f\left(x\right)=4[/latex].
38. Given the function [latex]f\left(x\right)=\sqrt{x+2}:[/latex]
a. Evaluate [latex]f\left(7\right)[/latex].
b. Solve [latex]f\left(x\right)=4[/latex].
39. Consider the relationship [latex]3r+2t=18[/latex].
a. Write the relationship as a function [latex]r=f\left(t\right)[/latex].
b. Evaluate [latex]f\left(-3\right)[/latex].
c. Solve [latex]f\left(t\right)=2[/latex].
For the following exercises, use the vertical line test to determine which graphs show relations that are functions.
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52. Given the following graph,
a. Evaluate [latex]f\left(-1\right)[/latex].
b. Solve for [latex]f\left(x\right)=3[/latex].
53. Given the following graph,
a. Evaluate [latex]f\left(0\right)[/latex].
b. Solve for [latex]f\left(x\right)=-3[/latex].
54. Given the following graph,
a. Evaluate [latex]f\left(4\right)[/latex].
b. Solve for [latex]f\left(x\right)=1[/latex].
For the following exercises, determine if the given graph is a one-to-one function.
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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]y[/latex] as a function of [latex]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]f[/latex] represented in the table below.
| [latex]x[/latex] | [latex]f\left(x\right)[/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\left(3\right)[/latex].
67. Solve [latex]f\left(x\right)=1[/latex].
For the following exercises, evaluate the function [latex]f[/latex] at the values [latex]f\left(-2\right),f\left(-1\right),f\left(0\right),f\left(1\right)[/latex], and [latex]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,g[/latex], and [latex]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]
Applications
76. The amount of garbage, [latex]G[/latex], produced by a city with population [latex]p[/latex] is given by [latex]G=f\left(p\right)[/latex]. [latex]G[/latex] is measured in tons per week, and [latex]p[/latex] 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 [latex]f[/latex].
b. Explain the meaning of the statement [latex]f\left(5\right)=2[/latex].
77. The number of cubic yards of dirt, [latex]D[/latex], needed to cover a garden with area [latex]a[/latex] square feet is given by [latex]D=g\left(a\right)[/latex].
a. A garden with area 5000 ft2 requires 50 yd3 of dirt. Express this information in terms of the function [latex]g[/latex].
b. Explain the meaning of the statement [latex]g\left(100\right)=1[/latex].
78. Let [latex]f\left(t\right)[/latex] be the number of ducks in a lake [latex]t[/latex] years after 1990. Explain the meaning of each statement:
a. [latex]f\left(5\right)=30[/latex]
b. [latex]f\left(10\right)=40[/latex]
79. Let [latex]h\left(t\right)[/latex] be the height above ground, in feet, of a rocket [latex]t[/latex] seconds after launching. Explain the meaning of each statement:
a. [latex]h\left(1\right)=200[/latex]
b. [latex]h\left(2\right)=350[/latex]
80. Show that the function [latex]f\left(x\right)=3{\left(x - 5\right)}^{2}+7[/latex] is not one-to-one.
See the next page for the solutions to the odd-numbered problems.