MATH 1301 (College Algebra) - Final Exam Review

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1 MATH 1301 (College Algebra) - Final Exam Review This review is comprehensive but should not be the only material used to study for the final exam. It should not be considered a preview of the final exam. Studying previous tests, quizzes, homework, class notes, text discussions, etc. will prepare a student to do well on the final exam. There may be questions on the final exam unlike questions on this review, and vice versa. No question on this review will be exactly duplicated on the final exam. This review is longer than the final exam. You may obtain help working on this review in the Math Lab located in 925-N. 1. Classify each number. Choose all correct answers. a. È * : (i) natural number (ii) integer (iii) rational number (iv) real number b. È : (i) natural number (ii) integer (iii) rational number (iv) real number c. : (i) natural number (ii) integer (iii) rational number (iv) real number ( d.! : (i) natural number (ii) integer (iii) rational number (iv) real number 2. For each measured quantity, choose the set of numbers that is most appropriate to describe it. (There is one correct answer for each part.) a. Population of a town: (i) natural number (ii) integer (iii) rational number (iv) real number b. Gallons of gasoline purchased at last fill-up: (i) natural number (ii) integer (iii) rational number (iv) real number 3. Convert the number as indicated. i. Convert to scientific notation:!þ!!!!!!!) ii. Convert to standard notation: %Þ! ' 4. The following table shows the total enrollment of students at Texas Southern University (TSU) from 2000 to (years after 2000)! $ % ' ( C (enrollment in!!!'s) 'Þ) )Þ! *Þ)!Þ* Þ Þ% Þ$ *Þ Find the percent change in the enrollment at TSU from 2003 to Interpret your answer. A. From 2003 to 2007, the change in TSU's enrollment was Þ%.. From 2003 to 2007, the percent change in TSU's enrollment was!þ). C. From 2003 to 2007, TSU's enrollment decreased by Þ)%. D. From 2003 to 2007, TSU's enrollment decreased by %Þ(%. 5. Which of the following graphs are functions? i. ii. iii. A. i) only. iii) only C. i) and iii) D. i) and ii) 1

2 6. Which of the following graphs are not functions? i. ii. iii. iv. 7. Which of the following relations are functions? i.! $ ii.!! $ iii.! $ iv.! C! $ C ( C C! ) ( ( 8. What is the domain of the function 0ÐÑ œ? 9. What is the domain of the function CœÈ '? A. '. ' C. Ÿ! D. Ÿ' E.! 10. What is the domain of the function 0ÐÑ œ? Ð 'ÑÐ Ñ 11. What is the domain of the function 0ÐÑ œ $? 12. Find the domain and range of the function 0 graphed here. A. domain œ Ò!ß $Óß range œ Ò ß Ó. domain œ Ò!ß $Óß range œò ßÓ C. domain œò ßÓßrange œò!ß$ó D. domain œò ß%Óßrange œò $ß'Ó E. domain œð ß Ñßrange œð ß Ñ 13 Þ For the function 0 graphed in Problem 12, determine: i. 0Ð!Ñß ii. 0ÐÑß iii. 0ÐÑß iv. 0Ð$Ñ 14. For the function 2 graphed here, determine: i. 2Ð Ñ ii Þ2Ð!Ñ iii. 2ÐÑ 15. Given that 0ÐÑ œ $, find 0Ð+ ÑÞ A. $+ %. $+ C. $+ D. $+ ) E. $+ $ 16. Fill in the blanks: If ;Ð ÞÑ œ $Þ, then the point Ð ß Ñ lies on the graph of ;. 2

3 17. For a person inches tall with a sprained ankle, the length of crutches, in inches, that he or she may need is approximated by 0ÐÑ œ!þ*. Evaluate and interpret 0Ð(Ñ. A. A person ( inches tall needs crutches that are approximately ( inches long.. A person ( inches tall needs crutches that are approximately ( inches long. C. A person ( inches tall needs crutches that are approximately %' inches long. D. A person ( inches tall needs crutches that are approximately ( inches long. 18. Which of the following lines have negative slope? i. ii. iii. iv. 19. Find the slope and the C -intercept of the line: $C $ œ Which of the following is not true about the graph of the line % C œ!? A. It rises from left to right.. It has C-intercept œ %. C. It passes through the point Ð!ß %Ñ. D. It has slope. E. It has -intercept œ. % 21. The slope of the line $ C œ 0 is: $ A.. C. D. E. $ $ $ 22. The slope of the line passing through the points Ð 3ß 1 Ñ and Ðß 3 Ñ is: A.. C. D. E. undefined % 23. The slope of a line which is perpendicular to the line Cœ ' is: A.. C. D. E. ' 24. The equation of the line passing through the points Ðß Ñ and Ð ß $Ñ is: A. Cœ $. Cœ C. Cœ D. Cœ E. Cœ 25. Find the equation of the line with -intercept œ and C-intercept œ $. 26. Which of the following is the equation of the line passing through the point Ð ß $Ñ and perpendicular to the line $ C œ? A. C $ œ. $ C œ $ C. $C œ D. $C œ ( E. $C œ ( 27. Which of the following is true for the lines with equations Cœ and Cœ? A. The lines are parallel Þ. The lines are perpendicularþ C. The lines are neither parallel nor perpendicular Þ D. The lines are the sameþ E. The lines are horizontal. 3

4 28. The equation of the line with slope passing through the point Ð ß $Ñ is: A. C œ (. C œ ( C. C œ $ D. C œ $ E. C œ ( 29. Determine if the function 0 shown here is linear or nonlinear. ' ( ) 0ÐÑ $ %! % 30. The correct dosage of a particular drug is! mg plus mg for each pound the patient weighs. Let C be the correct dosage for a patient weighing pounds. Write a formula for this function. A. œ!c.! C œ! C. C œ! D. C œ! E.!C œ 31. A particular car that costs $ ß!!! new is projected to lose $ $!! in resale value each month. Let Z be the resale value after > months. Write a formula for this function. 32Þ The typical weight C, in pounds, for a healthy baby elephant that is months old is given by C œ 0ÐÑ œ '!! i. Interpret the meaning of the expression 0ÐÑ. Do not evaluate. ii. Find the C-intercept of this line and interpret its meaning. iii. Find the slope of this line and interpret its meaning. 33. The amount of oil produced in the U.S. has been decreasing since about Suppose the following linear function relates the amount of oil, C, produced daily in the U.S. (measured in millions of barrels) to the number of years,, since C œ 0ÐÑ œ!!þ i. Interpret the meaning of the expression 0Ð$(Ñ. Do not evaluate. ii. Find the -intercept of this line and interpret its meaning. iii. Find the slope of this line and interpret its meaning. 34. The cost of coffee is directly proportional to the amount purchased. Two and a half pounds of coffee cost $. How many pounds of coffee can be bought with $? 35. The quantity C is directly proportional to the quantity, and C œ when œ $. Find the constant of proportionality 5. A. 5œ. 5œ C. 5œ$ D. 5œ $ 36. Which of the following intervals is the solution set of $ %? A. Ð ß )Ñ. Ð )ß Ñ C. Ð ß Ñ D. Ð ß Ñ E. Ð ß Ñ 37. Solve each inequality, graph the solution set, and write the answer in interval notation. i. $ ( ii. Ÿ $ 38. Solve the compound inequality, graph the solution set, and write the answer in interval notation. $ Ÿ Ÿ $ 39. Evaluate 0ÐÑ given that $ ßif! 0ÐÑ œ œ È ß if! A.!. È$ C. D. 4

5 40. Using the piecewise function defined in Problem 39, evaluate the following: i. 0Ð $Ñ ii. 0Ð!Ñ iii. 0Ð'Ñ 41. The graph of a function 0 is shown here. i. Where is 0ÐÑ!? ii. Where is 0ÐÑ Ÿ!? 42 Þ For the function 0 graphed in Problem 41, find all values for such that 0ÐÑ œ!. % $ 43. The graph of the function 0ÐÑ œ $ is shown here. % $ Use this graph to approximate a solution to the equation $ œ!. A.!Þ(.! C. Þ% D. E. 44. Solve: l (l œ 45 Þ Solve the inequality and write the solution in interval notation: l (l 46. The vertex of the parabola with equation Cœ% ) ( is: A. Ðß (Ñ. Ðß $Ñ C. Ð ß $Ñ D. Ð ß (Ñ E. Ð%ß (Ñ 47. The -intercepts of the parabola with equation C œ ) ( are: A. )ß(. ß) C. (ß D. ß( E. it has none 48. Which of the following is true for the graph of the function 0ÐÑ œ %? A. It has vertex at Ð ß Ñ and opens downward.. It has vertex at Ðß $Ñ and opens downward. C. It has vertex at Ð ß $Ñ and opens upward. D. It has vertex at Ðß $Ñ and opens upward. E. It has vertex at Ðß ÑÞ 5

6 49. Which of the following graphs represents the function Cœ %? A.. C. D. 50. Solve the equation: % œ Solve the equation: œ ( 52. Solve the equation: CÐC $Ñ œ! 53. Solve the equation: A 'A $ œ Solve the equation: œ % % È%3 % 3 È% % 3 È% A.. C. % % % % È' % 3 È' D. E. % % 55 Þ The height of an object tossed upward with an initial velocity of! feet per second is 2 given by the formula 2 œ '>!>, where 2 is the height in feet and > is the time in seconds. Find the time required for the object to return to its point of departure. 56 Þ The height of an object tossed upward with an initial velocity of '! feet per second is 2 given by the formula 2 œ '> '!>, where 2 is the height in feet and > is the time in seconds. Find the maximum height of the object. A. %!!. )! C. '! D. %% 6

7 57 Þ Solve the inequality and write the solution in interval notation:! 58 Þ Solve the inequality and write the approximate solution in interval notation: %! A. Ð ß Ó. Ð ß ÞÑ Ð$Þß Ñ C. Ò%ß Ñ D. Ð Þß $ÞÑ 59. Which one of the following functions is a polynomial? What is its C-intercept? What is its domain? $ A. 0ÐÑ œ % $. 0ÐÑ œ È C. 0ÐÑ œ $ $ D. 0ÐÑ œ $ % E. 0ÐÑ œ $ 60. What is the degree of the polynomial function 0ÐÑ œ %!Þ? What is the leading coefficient? 61. At most how many -intercepts can a degree polynomial function have? A. %. $ C. D. infinitely many % th 62. At most how many turning points can a degree polynomial function have? A. %. $ C. D. infinitely many % th 7

8 63. The graph of a polynomial function is shown here. What is the minimum degree this polynomial could have? A.!. C. D. $ 64. Which of the following points is a turning point of the function graphed in Problem 63? A. Ð %ß Ñ. Ð$Þß!Ñ C. Ð ß %Ñ D. Ð!ß!Ñ E. Ð $Þß!Ñ 65. Which of the following values is a local minimum of the function graphed in Problem 63? A.. % C. D. % 66. Use the graph in Problem 63 to determine the set where the function is decreasing. Use interval notation. A. Ò $Þß!Ó. Ð ß Ó Òß Ñ C. Ò%ß %Ó D. Ò ß Ó 67. Where is the function 1ÐÑ œ % % increasing? % 68. Determine the end behavior of the graph of 0ÐÑ œ. A. It rises on the left and rises on the right.. It falls on the left and falls on the right. C. It falls on the left and rises on the right. D. It rises on the left and falls on the right. $ 69. Determine the end behavior of the graph of 0ÐÑ œ!þ %. 70. Which one of the following graphs could be a polynomial function? i. ii. iii. iv. 71. Write each expression in +,3 form. i. Ð$ È $' Ñ Ð È ' Ñ ii. Ð$ %3ÑÐ 3Ñ 72. Simplify: Ð 3 Ñ 73. Let 0ÐÑ œ and 1ÐÑ œ. Then Ð0 1ÑÐÑ œ A. % $. 1 C. D. Ð Ñ E. Ð ÑÐ Ñ 74. Let 0ÐÑ œ È and 1ÐÑ œ $. Find 0Ð1ÐÑÑ Þ A. %. C. D. È$ E. È$ 8

9 75. For the functions 0 and 1 graphed here, evaluate Ð0 1ÑÐ%Ñ Which one of the following statements is false? A. Every one-to-one function has an inverse function.. The graph of a function and its inverse are reflections of each other about the line Cœ. C. The graph of a one-to-one function passes the horizontal line test. D. If 0 is a function, its inverse function is Î0Þ E. The composition of a function and its inverse is the identity function. 77. Let 0ÐÑ œ $ 'Þ Then 0 ÐÑ œ ' A.. $ ' C. D. E. ' $ $ ' $ $ 78. Let 0ÐÑ œ È Þ Then 0 ÐÑ œ A. È. C. È D. E. È 79. For a person inches tall with a sprained ankle, the length of crutches, in inches, that he or she may need is approximated by 0ÐÑ œ!þ*. Interpret the meaning of 0 Ð%)Ñ œ!. A. Crutches that are! inches long are appropriate for a person %) inches tall.. Crutches that are! inches long are appropriate for a person *) inches tall. C. Crutches that are %) inches long are appropriate for a person %) inches tall. D. Crutches that are %) inches long are appropriate for a person! inches tall. 80. Is the function 1 shown here exponential?! $ 1ÐÑ!! %! 81. The intensity M of light (in lumens) meters below the water surface is given by the formula M œ M 5, where M! is the intensity of light on the water surface and 5 is the clarity constant of the water. At one location in the Atlantic Ocean, M! œ! and 5 œ!þ'. Find the intensity of light at a depth of meters. A. lumens. ' lumens C. $Þ' lumens D.!Þ$ lumens E. Þ lumens 82. For the exponential function 0ÐÑ œ $ÐÞÑ, evaluate 0Ð Ñ. A. %Þ.!Þ C. Þ D. 83. For the exponential function 0ÐÑ œ %, evaluate 0Š.! 9

10 84. Graph the function: Cœ%. Which one of the following statements is false? A. domain œ Ð ß Ñ. range œ Ð!ß Ñ C. The C-intercept is Ð0, 1Ñ D. The graph passes through the point Ð, 'Ñ E. No -intercept F. The graph passes through the point Ð ß Ñ % 85. Which of the following graphs could represent the function 0ÐÑ œ Š? $ A.. C. D. 86. Which of the graphs in Problem 85 could represent the function 0ÐÑ œ log? 87. Which of the following functions is graphed here? A. Cœ. Cœ% C. Cœ D. Cœ 88. Which equation is equivalent to $ œ? * A. log Ð Ñ œ. log $ œ C. log œ $ $ * * * D. log Ð Ñ * œ$ Î 89. Express in logarithmic form: i. œ ii. Œ œ$ 90. Write the following equations in exponential form: i. log $œ ii. log œ$ 10

11 91. Evaluate each of the following: i. log $ ( ii. log iii. log iv. ln! 92. Evaluate: log ) A. %. ' C. $ D. $ 93. Find the value of if log $ œ Þ A. œ. œ* C. œ D. œ ' ) * 94. Find the value of if ln œ. A. œ/. œ/ C. œ! D. œ! 95. Find the value of if log $' œ Þ A. œ'. œ) C. œ D. œ ) ' ) 96. Rewrite the following as a single logarithmic expression: ln ln 97. Rewrite the following as a single logarithmic expression: log Ð Ñ log Ð Ñ $ $ 98. Solve:! œ! A. œ. œ log! C. œ! log! D. œlog E. œ!! 99. Solve: / œ! ANSWERS 1a. i,ii,iii,iv 1b. iv 1c. iii,iv 1d. ii,iii,iv 2a. i 2b. iii 3i. Þ)! ) 3ii. %!!!! 4. C 5. D 6. iii,iv 7. i,iii,iv 8. All real numbers except Î 9. A 10. All real numbers except and ' 11. All real numbers i. 13ii. 13iii.! 13iv. 14i. Þ( 14ii.! 14iii. Þ( 15. C 16. Ð Þß $ÞÑ 17. A 18. i,iv 19. slope œ, C-intercept œ $ 20. D 21. A 22. C 23. $ 24. D 25. Cœ $ 26. C 27. A nonlinear 30. D 31. Z œ!!! $!!> 32i. 0ÐÑ is the typical weight of a healthy baby elephant that is months old. 32ii. C-intercept œ!; A healthy newborn baby elephant typically weighs! lb. 32iii. slope œ'! ; A healthy baby elephant typically gains '! lb per month. 33i. 0Ð$(Ñ is the amount of oil produced daily in the U.S. in ii. -intercept œ )!; y 2050, no oil will be produced in the U.S. 33iii. slope œ!þ ; Since 1970, the amount of oil produced daily in the U.S. has decreased by!þ million barrels per year. 34. lb E 11

12 37i., Ð 'ß Ñ 37ii., Ò%ß Ñ 38., Ò, )Ó 39. D 40i. ) 40ii. È$ 40iii. $ 41i. Ð $Þß!Ñ Ð$Þß Ñ 41ii. Ð ß $ÞÓ Ò!ß $ÞÓ 42. œ $Þß!ß $Þ * * 43. C 44. œß 45. Ðß Ñ 46. C % È 47. C C 50. œ ( 51. œ ß 52. Cœ ß 53. Aœ $ C 55. (Þ sec 56. A 57. Ð ß $Ó Ò%ß Ñ 58. D 59. A; C-intercept œ$ ; domain œall real numbers 60. degree œ ; leading coefficient œ!þ 61. A D 64. C D 67. Ð ß Ó 68. A 69. The graph rises on the left and falls on the right. 70. iii 71i. 3 71ii Þ A 74. C 75.! 76. D 77. C 78. D 79. D 80. Yes 81. C 82. D D 85. A A 88. C 89i. log œ $ 89ii. log $œ 90i. œ$ 90ii. œ 91i. $ 91ii. 91iii. 91iv.! 92. C 93. C A 96. ln ' 97. log È $ œ ln! 12