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学習ガイド > College Algebra

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. {(a,b), (c,d), (a,c)}\left\{\left(a,b\right),\text{ }\left(c,d\right),\text{ }\left(a,c\right)\right\} 7. {(a,b),(b,c),(c,c)}\left\{\left(a,b\right),\left(b,c\right),\left(c,c\right)\right\}   For the following exercises, determine whether the relation represents yy as a function of xx. 8. 5x+2y=105x+2y=10 9. y=x2y={x}^{2} 10. x=y2x={y}^{2} 11. 3x2+y=143{x}^{2}+y=14 12. 2x+y2=62x+{y}^{2}=6 13. y=−2x2+40xy=-2{x}^{2}+40x 14. y=1xy=\frac{1}{x} 15. x=3y+57y−1x=\frac{3y+5}{7y - 1} 16. x=1−y2x=\sqrt{1-{y}^{2}} 17. y=3x+57x−1y=\frac{3x+5}{7x - 1} 18. x2+y2=9{x}^{2}+{y}^{2}=9 19. 2xy=12xy=1 20. x=y3x={y}^{3} 21. y=x3y={x}^{3} 22. y=1−x2y=\sqrt{1-{x}^{2}} 23. x=±1−yx=\pm \sqrt{1-y} 24. y=±1−xy=\pm \sqrt{1-x} 25. y2=x2{y}^{2}={x}^{2} 26. y3=x2{y}^{3}={x}^{2} For the following exercises, evaluate the function ff at the indicated values  f(−3),f(2),f(−a),−f(a),f(a+h)\text{ }f\left(-3\right),f\left(2\right),f\left(-a\right),-f\left(a\right),f\left(a+h\right). 27. f(x)=2x−5f\left(x\right)=2x - 5 28. f(x)=−5x2+2x−1f\left(x\right)=-5{x}^{2}+2x - 1 29. f(x)=2−x+5f\left(x\right)=\sqrt{2-x}+5 30. f(x)=6x−15x+2f\left(x\right)=\frac{6x - 1}{5x+2} 31. f(x)=∣x−1∣−∣x+1∣f\left(x\right)=|x - 1|-|x+1| 32. Given the function g(x)=5−x2g\left(x\right)=5-{x}^{2}, evaluate g(x+h)−g(x)h,h≠0\frac{g\left(x+h\right)-g\left(x\right)}{h},h\ne 0. 33. Given the function g(x)=x2+2xg\left(x\right)={x}^{2}+2x, evaluate g(x)−g(a)x−a,x≠a\frac{g\left(x\right)-g\left(a\right)}{x-a},x\ne a. 34. Given the function k(t)=2t−1:k\left(t\right)=2t - 1:

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

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

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

36. Given the function p(c)=c2+c:p\left(c\right)={c}^{2}+c:

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

37. Given the function f(x)=x2−3x:f\left(x\right)={x}^{2}-3x:

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

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

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

39. Consider the relationship 3r+2t=183r+2t=18.

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

For the following exercises, use the vertical line test to determine which graphs show relations that are functions. 40. Graph of relation. 41. Graph of relation. 42. Graph of relation. 43. Graph of relation. 44. Graph of relation. 45. Graph of relation. 46. Graph of relation. 47. Graph of relation. 48. Graph of relation. 49. Graph of relation. 50. Graph of relation. 51. Graph of relation. 52. Given the following graph,

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

Graph of relation. 53. Given the following graph,

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

Graph of relation. 54. Given the following graph,

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

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. {(−1,−1),(−2,−2),(−3,−3)}\left\{\left(-1,-1\right),\left(-2,-2\right),\left(-3,-3\right)\right\} 61. {(3,4),(4,5),(5,6)}\left\{\left(3,4\right),\left(4,5\right),\left(5,6\right)\right\} 62. {(2,5),(7,11),(15,8),(7,9)}\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 yy as a function of xx. 63.
xx 5 10 15
yy 3 8 14
64.
xx 5 10 15
yy 3 8 8
65.
xx 5 10 10
yy 3 8 14
For the following exercises, use the function ff represented in the table below.
xx f(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(3)f\left(3\right). 67. Solve f(x)=1f\left(x\right)=1. For the following exercises, evaluate the function ff at the values f(−2),f(−1),f(0),f(1)f\left(-2\right),f\left(-1\right),f\left(0\right),f\left(1\right), and f(2)f\left(2\right). 68. f(x)=4−2xf\left(x\right)=4 - 2x 69. f(x)=8−3xf\left(x\right)=8 - 3x 70. f(x)=8x2−7x+3f\left(x\right)=8{x}^{2}-7x+3 71. f(x)=3+x+3f\left(x\right)=3+\sqrt{x+3} 72. f(x)=x−2x+3f\left(x\right)=\frac{x - 2}{x+3} 73. f(x)=3xf\left(x\right)={3}^{x} For the following exercises, evaluate the expressions, given functions f,gf,g, and h:h:
  • f(x)=3x−2f\left(x\right)=3x - 2
  • g(x)=5−x2g\left(x\right)=5-{x}^{2}
  • h(x)=−2x2+3x−1h\left(x\right)=-2{x}^{2}+3x - 1
74. 3f(1)−4g(−2)3f\left(1\right)-4g\left(-2\right) 75. f(73)−h(−2)f\left(\frac{7}{3}\right)-h\left(-2\right) For the following exercises, graph y=x2y={x}^{2} on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. 76. [−0.1, 0.1]\left[-0.1,\text{ }0.1\right] 77. [−10, 10]\left[-10,\text{ 10}\right] 78. [−100,100]\left[-100,100\right] For the following exercises, graph y=x3y={x}^{3} on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. 79. [−0.1, 0.1]\left[-0.1,\text{ 0}\text{.1}\right] 80. [−10, 10]\left[-10,\text{ 10}\right] 81. [−100, 100]\left[-100,\text{ 100}\right] For the following exercises, graph y=xy=\sqrt{x} on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. 82. [0, 0.01]\left[0,\text{ 0}\text{.01}\right] 83. [0, 100]\left[0,\text{ 100}\right] 84. [0, 10,000]\left[0,\text{ 10,000}\right] For the following exercises, graph y=x3y=\sqrt[3]{x} on the given viewing window. Determine the corresponding range for each viewing window. Show each graph. 85. [−0.001,0.001]\left[-0.001,\text{0.001}\right] 86. [−1000,1000]\left[-1000,\text{1000}\right] 87. [−1,000,000,1,000,000]\left[-1,000,000,\text{1,000,000}\right] 88. The amount of garbage, GG, produced by a city with population pp is given by G=f(p)G=f\left(p\right). GG is measured in tons per week, and pp 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 ff. b. Explain the meaning of the statement f(5)=2f\left(5\right)=2.

89. The number of cubic yards of dirt, DD, needed to cover a garden with area aa square feet is given by D=g(a)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 gg. b. Explain the meaning of the statement g(100)=1g\left(100\right)=1.

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

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

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

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

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

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