{"id":10767,"date":"2015-07-10T19:40:02","date_gmt":"2015-07-10T19:40:02","guid":{"rendered":"https:\/\/courses.candelalearning.com\/osprecalc\/?post_type=chapter&p=10767"},"modified":"2018-06-18T19:34:35","modified_gmt":"2018-06-18T19:34:35","slug":"solutions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ccbcmd-math\/chapter\/solutions\/","title":{"raw":"Solutions 2.1: Functions and Function Notation","rendered":"Solutions 2.1: Functions and Function Notation"},"content":{"raw":"

Solutions to Try Its<\/span><\/h2>\r\n1. a. yes; b. yes.\u00a0(Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)\r\n\r\n2.\u00a0[latex]g\\left(5\\right)=1[\/latex]\r\n\r\n3. [latex]m=8[\/latex]\r\n\r\n4. [latex]y=f\\left(x\\right)=\\frac{\\sqrt[3]{x}}{2}[\/latex]\r\n\r\n5. [latex]g\\left(1\\right)=8[\/latex]\r\n\r\n6. [latex]x=0[\/latex] or [latex]x=2[\/latex]\r\n\r\n7.\u00a0a. yes, because each bank account has a single balance at any given time;\u00a0<\/span>b. no, because several bank account numbers may have the same balance;\u00a0<\/span>c. no, because the same output may correspond to more than one input.<\/span>\r\n\r\n8. yes\r\n\r\n \r\n

Solutions to Odd-Numbered Exercises<\/span><\/h2>\r\n1. 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.\r\n\r\n3. 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.\r\n\r\n5. 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.\r\n\r\n7. function\r\n\r\n9. function\r\n\r\n11. function\r\n\r\n13. function\r\n\r\n15. function\r\n\r\n17. function\r\n\r\n19. function\r\n\r\n21. function\r\n\r\n23. function\r\n\r\n25. not a function\r\n\r\n27. [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]\r\n\r\n29. [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]\r\n\r\n31. [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]\r\n\r\n33. [latex]\\frac{g\\left(x\\right)-g\\left(a\\right)}{x-a}=x+a+2,x\\ne a[\/latex]\r\n\r\n35. a. [latex]f\\left(-2\\right)=14[\/latex]; b. [latex]x=3[\/latex]\r\n\r\n37. a. [latex]f\\left(5\\right)=10[\/latex]; b. [latex]x=-1\\text{ }[\/latex] or [latex]\\text{ }x=4[\/latex]\r\n\r\n39. 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]\r\n\r\n41. not a function\r\n\r\n43. function\r\n\r\n45. function\r\n\r\n47. function\r\n\r\n49. function\r\n\r\n51. function\r\n\r\n53. 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]\r\n\r\n55. not a function so it is also not a one-to-one function\r\n\r\n57. one-to-one function\r\n\r\n59. function, but not one-to-one\r\n\r\n61. function\r\n\r\n63. function\r\n\r\n65. not a function\r\n\r\n67. [latex]f\\left(x\\right)=1,x=2[\/latex]\r\n\r\n69. [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]\r\n\r\n71. [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]\r\n\r\n73. [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]\r\n\r\n75. 20\r\n\r\n77. [latex]\\left[0,\\text{ 100}\\right][\/latex]\r\n\r\n\"Graph\r\n\r\n79.[latex]\\left[-0.001,\\text{ 0}\\text{.001}\\right][\/latex]\r\n\r\n\"Graph\r\n\r\n81. [latex]\\left[-1,000,000,\\text{ 1,000,000}\\right][\/latex]\r\n\r\n\"Graph\r\n\r\n83. [latex]\\left[0,\\text{ 10}\\right][\/latex]\r\n\r\n\"Graph\r\n\r\n85.\u00a0[latex]\\left[-0.1,\\text{0.1}\\right][\/latex]\r\n\r\n\"Graph\r\n\r\n87. [latex]\\left[-100,\\text{ 100}\\right][\/latex]\r\n\r\n\"Graph\r\n\r\n89.\u00a0a. [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.\r\n\r\n91. 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.","rendered":"

Solutions to Try Its<\/span><\/h2>\n

1. a. yes; b. yes.\u00a0(Note: If two players had been tied for, say, 4th place, then the name would not have been a function of rank.)<\/p>\n

2.\u00a0[latex]g\\left(5\\right)=1[\/latex]<\/p>\n

3. [latex]m=8[\/latex]<\/p>\n

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

5. [latex]g\\left(1\\right)=8[\/latex]<\/p>\n

6. [latex]x=0[\/latex] or [latex]x=2[\/latex]<\/p>\n

7.\u00a0a. yes, because each bank account has a single balance at any given time;\u00a0<\/span>b. no, because several bank account numbers may have the same balance;\u00a0<\/span>c. no, because the same output may correspond to more than one input.<\/span><\/p>\n

8. yes<\/p>\n

 <\/p>\n

Solutions to Odd-Numbered Exercises<\/span><\/h2>\n

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.<\/p>\n

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.<\/p>\n

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.<\/p>\n

7. function<\/p>\n

9. function<\/p>\n

11. function<\/p>\n

13. function<\/p>\n

15. function<\/p>\n

17. function<\/p>\n

19. function<\/p>\n

21. function<\/p>\n

23. function<\/p>\n

25. not a function<\/p>\n

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

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

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

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

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]<\/p>\n

41. not a function<\/p>\n

43. function<\/p>\n

45. function<\/p>\n

47. function<\/p>\n

49. function<\/p>\n

51. function<\/p>\n

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]<\/p>\n

55. not a function so it is also not a one-to-one function<\/p>\n

57. one-to-one function<\/p>\n

59. function, but not one-to-one<\/p>\n

61. function<\/p>\n

63. function<\/p>\n

65. not a function<\/p>\n

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

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]<\/p>\n

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]<\/p>\n

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]<\/p>\n

75. 20<\/p>\n

77. [latex]\\left[0,\\text{ 100}\\right][\/latex]<\/p>\n

\"Graph<\/p>\n

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

\"Graph<\/p>\n

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

\"Graph<\/p>\n

83. [latex]\\left[0,\\text{ 10}\\right][\/latex]<\/p>\n

\"Graph<\/p>\n

85.\u00a0[latex]\\left[-0.1,\\text{0.1}\\right][\/latex]<\/p>\n

\"Graph<\/p>\n

87. [latex]\\left[-100,\\text{ 100}\\right][\/latex]<\/p>\n

\"Graph<\/p>\n

89.\u00a0a. [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.<\/p>\n

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.<\/p>\n\n\t\t\t

\n\t\t\t

Candela Citations<\/h3>\n\t\t\t\t\t
\n\t\t\t\t\t\t
\n\t\t\t\t\t\t\t
CC licensed content, Shared previously<\/div>