Solutions 1: Functions and Function Notation

Solutions to Try Its

1. a. yes; b. yes. (Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)

2. [latex]g\left(5\right)=1[/latex]

3. [latex]m=8[/latex]

4. [latex]y=f\left(x\right)=\frac{\sqrt[3]{x}}{2}[/latex]

5. [latex]g\left(1\right)=8[/latex]

6. [latex]x=0[/latex] or [latex]x=2[/latex]

7. a. yes, because each bank account has a single balance at any given time; b. no, because several bank account numbers may have the same balance; c. no, because the same output may correspond to more than one input.

8. yes

 

Solutions to Odd-Numbered Exercises

1. A relation is a set of ordered pairs. A function is a special kind of relation in which no two ordered pairs have the same first coordinate.

3. When a vertical line intersects the graph of a relation more than once, that indicates that for that input there is more than one output. At any particular input value, there can be only one output if the relation is to be a function.

5. When a horizontal line intersects the graph of a function more than once, that indicates that for that output there is more than one input. A function is one-to-one if each output corresponds to only one input.

7. function

9. function

11. function

13. function

15. function

17. function

19. function

21. function

23. function

25. not a function

27. [latex]f\left(-3\right)=-11\\ f\left(2\right)=-1\\ f\left(-a\right)=-2a - 5\\-f\left(a\right)=-2a+5\\f\left(a+h\right)=2a+2h - 5[/latex]

29. [latex]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\\f\left(a+h\right)=\\sqrt{2-a-h}+5[/latex]

31. [latex]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|+|a+1|\\\text{ }\text{ }f\left(a+h\right)=|a+h - 1|-|a+h+1|[/latex]

33. [latex]\frac{g\left(x\right)-g\left(a\right)}{x-a}=x+a+2,x\ne a[/latex]

35. a. [latex]f\left(-2\right)=14[/latex]; b. [latex]x=3[/latex]

37. a. [latex]f\left(5\right)=10[/latex]; b. [latex]x=-1\text{ }[/latex] or [latex]\text{ }x=4[/latex]

39. a. [latex]f\left(t\right)=6-\frac{2}{3}t[/latex]; b. [latex]f\left(-3\right)=8[/latex]; c. [latex]t=6[/latex]

41. not a function

43. function

45. function

47. function

49. function

51. function

53. a. [latex]f\left(0\right)=1[/latex]; b. [latex]f\left(x\right)=-3,x=-2\text{ }[/latex] or [latex]\text{ }x=2[/latex]

55. not a function so it is also not a one-to-one function

57. one-to-one function

59. function, but not one-to-one

61. function

63. function

65. not a function

67. [latex]f\left(x\right)=1,x=2[/latex]

69. [latex]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[/latex]

71. [latex]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[/latex]

73. [latex]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[/latex]

75. 20

77. [latex]\left[0,\text{ 100}\right][/latex]

Graph of a parabola.

79.[latex]\left[-0.001,\text{ 0}\text{.001}\right][/latex]

Graph of a cubic function.

81. [latex]\left[-1,000,000,\text{ 1,000,000}\right][/latex]

Graph of a cubic function.

83. [latex]\left[0,\text{ 10}\right][/latex]

Graph of a square root function.

85. [latex]\left[-0.1,\text{0.1}\right][/latex]

Graph of a cube root function.

87. [latex]\left[-100,\text{ 100}\right][/latex]

Graph of a cubic root function.

89. a. [latex]g\left(5000\right)=50[/latex]; b. The number of cubic yards of dirt required for a garden of 100 square feet is 1.

91. a. The height of a rocket above ground after 1 second is 200 ft. b. the height of a rocket above ground after 2 seconds is 350 ft.