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definition a b Β± c d ad Β± bc bd, b 0, d 0. Basic definition This definition provides an efficient way of adding or subtracting two fractions that have no common factors in their denominators. Example 6 Subtracting Rational Expressions x x 3 When subtracting rational expressions, remember to distribute the negative sig... |
fractions. Here are two examples. 1 x x 2 1 and 1 x 1 x 2 1 A complex fraction can be simplified by combining the fractions in its numerator into a single fraction and then combining the fractions in its denominator into a single fraction. Then invert the denominator and multiply. Example 8 Simplifying a Complex Fract... |
. 333202_0A04.qxd 12/6/05 2:16 PM Page A42 A42 Appendix A Review of Fundamental Concepts of Algebra Example 11 Rewriting a Difference Quotient The following expression from calculus is an example of a difference quotient. x h x h Rewrite this expression by rationalizing its numerator. Solution x2 hx h x h hx h x 1 x h ... |
5 6x 2 5 2x 9. 10. 3 4x 1 3 4 In Exercises 11β28, write the rational expression in simplest form. 12. 14. 16. 18. 20. 22. 24. 18y 2 60y 5 2x2y xy y 9x 2 9x 2x 2 12 4x x 3 x 2 25 5 x x 2 8x 20 x 2 11x 10 x2 7x 6 x 2 11x 10 11. 13. 15. 17. 19. 21. 23. 25. 26. 27. 28. 15x2 10x 3xy xy x 4y 8y2 10y 5 x 5 10 2x y2 16 y 4 x ... |
61β 66, factor the expression by removing the common factor with the smaller exponent. 43. 45. 47. 48. 49. 50. 51. 52 44. 46. 2x 2x 5x 6 10 x 2 2x Error Analysis In Exercises 53 and 54, describe the error. 53. x 4 x 2 3x 8 x 2 x 4 3x 8 x 2 2x 4 x 2 8 x 2x 2 54. 6 x xx 2 x 2 x 2 2x 2 x 2 2 x6 x x 22 8 x 2x 2 6x x 2 x 2... |
to respectively. What fractional part of the task has been completed? t5, and t Finance In Exercises 81 and 82, the formula that approximates the annual interest rate r of a monthly installment loan is given by r ] [24NM P N P NM 12 Appendix A.4 Rational Expressions A45 84. Interactive Money Management The table shows... |
projected number of households paying bills online to the projected number of households banking online. (d) Use the model from part (c) to find the ratio over the given years. Interpret your results. Synthesis is the temperature (in degrees Fahrenheit) and T where the time (in hours). t is True or False? the statemen... |
that is true for every real number in the domain of the variable is called an identity. The domain is the set of all real numbers for which the equation is defined. For example x2 9 x 3x 3 Identity is an identity because it is a true statement for any real value of x. The equation x 3x2 1 3x x 0, where Identity is an ... |
, you should check each solution in the original equation. For instance, you can check the solution to Example 1(a) as follows. 3x 6 0 32 6? 0 0 0 Write original equation. Substitute 2 for x. Solution checks. β Try checking the solution to Example 1(b). Example 1 Solving a Linear Equation a. b. 3x 6 0 3x 6 x 2 5x 4 3x ... |
is essential that you check your solutions. Example 3 An Equation with an Extraneous Solution Solve 1 x 2 3 x 2 6x x2 4. Solution The LCD is 1 x 2 x 2x 2. x 2 4, or x 2x 2 3 Multiply each term by this LCD. x 2x 2 6x x 2x 2 x2 4 x Β±2 In the original equation, extraneous solution, and the original equation has no soluti... |
Algebra Example 4 Solving a Quadratic Equation by Factoring a. 2x 2 9x 7 3 2x2 9x 4 0 2x 1x 4 0 2x Original equation Write in general form. Factor. Set 1st factor equal to 0. Set 2nd factor equal to 0. b. The solutions are 6x 2 3x 0 3x2x 1 0 3x 0 2x 1 0 x 1 2 and x 4. Check these in the original equation. Original equ... |
6 12 half of 22 x 12 7 x 1 Β± 7 The solutions are x 1 Β± 7 x 1 Β± 7. Write original equation. Add 6 to each side. Add 12 to each side. Simplify. Take square root of each side. Subtract 1 from each side. Check these in the original equation. Now try Exercise 85. Example 7 Completing the Square: Leading Coefficient Is Not ... |
18 and 333202_0A05.qxd 12/6/05 2:45 PM Page A53 A common mistake that is made in solving an equation such as that in Example 10 is to divide each side of the equation by the x 2. variable factor This loses the x 0. solution When solving an equation, always write the equation in general form, then factor the equation a... |
an equation, raising each side of an equation to a rational power, and multiplying each side of an equation by a variable quantity all can introduce extraneous solutions. So, when you use any of these operations, checking your solutions is crucial. Example 12 Solving Equations Involving Radicals a. 2x 7 x 2 2x 7 x 2 2... |
inside the absolute value signs can be positive or negative, you must solve the following two equations. First Equation x2 3x 4x 6 x2 x 6 0 x 3x 2 0 x 3 0 x 2 0 Second Equation x2 3x 4x 6 x2 7x 6 0 x 1x 6 0 x 1 0 x 6 0 Check x 3 x 2 x 1 x 6 32 33? 43 6 18 18 22 32? 42 6 2 2 12 31? 41 6 2 2 62 36? 46 6 18 18 x 3 The so... |
6x 3 5 2x 10 3x 2 5 3x 1 4x 1 2x 2x 2 7x 3 4x 37 x x2 8x 5 x 42 11 x2 23x 2 x2 6x 4 4x 3 1 x 1 x 1 10. 5 x 3 x 24 In Exercises 11β26, solve the equation and check your solution. 12. 14. 16. 7 x 19 7x 2 23 7x 3 3x 17 22. x 5 x 2 3 3x 10 11. 13. 15. 17. 18. 19. 20. 21. 23 24. 25. 26. x 11 15 7 2x 25 8x 5 3x 20 2x 5 7 3x... |
. 67. 68. 6x 2 3x 0 x 2 2x 8 0 x 2 10x 25 0 3 5x 2x 2 0 x 2 4x 12 3 4 x2 8x 20 0 x 2 2ax a 2 0, x a2 b 2 0, a a 56. 58. 60. 62. 64. 66. 9x 2 1 0 x 2 10x 9 0 4x 2 12x 9 0 2x 2 19x 33 x 2 8x 12 1 8 x2 x 16 0 is a real number and are real numbers b In Exercises 69β82, solve the equation by extracting square roots. 69. 71.... |
3x x2 1 0 25h2 80h 61 0 z 62 2z 5 7x 142 8x In Exercises 117β124, use the Quadratic Formula to solve the equation. (Round your answer to three decimal places.) 117. 118. 119. 120. 121. 122. 123. 124. 5.1x 2 1.7x 3.2 0 2x 2 2.50x 0.42 0 0.067x2 0.852x 1.277 0 0.005x 2 0.101x 0.193 0 422x 2 506x 347 0 1100x2 326x 715 0 ... |
159. 161. 163. 165. 167. 169. 170. 171. 173. 175. 177. 179. 181. 183. 158. 160. 156. 154. 162. 2x 10 0 x 10 4 0 32x 5 3 0 26 11x 4 x x 1 3x 1 x 532 8 x 323 8 x2 532 27 168. 3xx 112 2x 132 0 4x2x 113 6xx 143 0 4 x 3 x x 1 1 2 166. 164. 172. 3 174. 1 x 20 2x 1 5 x x2 x 3 x 1 x 2 5 4x 3 0 5 x 3 0 33x 1 5 0 x 31 9x 5 x 5 ... |
feet of floor space. (a) Draw a diagram that gives a visual representation of and show the floor space. Represent the width as the length in terms of w. w y in. femur x in. (c) Find the length and width of the floor of the building. (b) Write a quadratic equation in terms of w. 189. Packaging An open box with a square... |
ribe the steps used to transform an equa- tion into an equivalent equation. E 203. To solve the equation 2 x2 3x 15x, each side by and solves the equation x 6 resulting solution Is there an error? Explain. a student divides 2x 3 15. The satisfies the original equation. x 193. Voting Population The total voting-age popu... |
b where and are constants. a 211. Find and when the solution of the equation is b a x 9. (There are many correct answers.) 212. Writing Write a short paragraph listing the steps required to solve this equation involving absolute values and explain why it is important to check your solutions. 213. Solve each equation, ... |
Solution 3, 5 a. 3, 0, 2, d. b. c. corresponds to corresponds to 3 < x β€ 5. 3 < x. 0 β€ x β€ 2. corresponds to corresponds to < x <. Now try Exercise 1. Bounded Unbounded Bounded Unbounded 333202_0A06.qxd 12/6/05 2:21 PM Page A61 Appendix A.6 Linear Inequalities in One Variable A61 Properties of Inequalities The procedu... |
is a linear inequality in x. In the following examples, pay special attention to the steps in which the inequality symbol is reversed. Remember that when you multiply or divide by a negative number, you must reverse the inequality symbol. Example 2 Solving Linear Inequalities Checking the solution set of an inequality... |
6x 1 < 3 3 1 β€ 6x 1 1 < 3 1 β€ 6x 6 2 β€ 6x < < Original inequality Add 1 to each part. Simplify. Divide each part by 6. Simplify. The solution set is all real numbers that are greater than or equal to. 3, 2 2 than which is denoted by 3, Figure A.10. and less The graph of this solution set is shown in 1 0 1 x Solution i... |
ve each inequality. x 5 < 2 a. b. x 3 β₯ 7 Notice that the graph is below the -axis on the interval 3, 7. x Solution a Write original inequality. Write equivalent inequalities. Add 5 to each part. Simplify. The solution set is all real numbers that are greater than 3 and less than 7, which is denoted by The graph of thi... |
dollars) Plan B costs more if you use more than 80 additional minutes in one month. Now try Exercise 91. Example 6 Accuracy of a Measurement You go to a candy store to buy chocolates that cost $9.89 per pound. The scale that is used in the store has a state seal of approval that indicates the scale is accurate to withi... |
the combining of two sets. In Exercises 1β 6, (a) write an inequality that represents the interval and (b) state whether the interval is bounded or unbounded. 1, 5 11,, 2 2, 10 5,, 7 1. 5. 3. 2. 6. 4. In Exercises 7β12, match the inequality with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).] (a)... |
.qxd 12/6/05 2:21 PM Page A67 39. 40. 41. 42. 43. 44. < 4 0 β€ 4 < 2x.2 β€ 0.4x 1 β€ 4.4 1.5x 6 2 4.5 > > 10.5 In Exercises 45β60, solve the inequality and sketch the solution on the real number line. (Some inequalities have no solution.) x < 6 x > 4 46. 45 20 β€ 6 x 8 β₯ 0 3 4x β₯ 9 1 2x < 5 2 β₯ 4 x 3 1 2x 3 < 1 47. 48. 49.... |
83β90, use absolute value notation to define the interval (or pair of intervals) on the real number line. 83. 84. 85. 86. β3 β2 β1 β3 β2 β 10 11 12 13 14 x x x x β7 β 6 β5 β4 β3 β2 β1 0 1 2 3 87. All real numbers within 10 units of 12 88. All real numbers at least five units from 8 89. All real numbers more than four ... |
greater than the cost. For what values of will this product return a profit? x 97. Daily Sales A doughnut shop sells a dozen doughnuts for $2.95. Beyond the fixed costs (rent, utilities, and insurance) of $150 per day, it costs $1.45 for enough materials (flour, sugar, and so on) and labor to produce a dozen doughnuts... |
οΏ½s maximum bench-press weight. 101. Teachersβ Salaries The average salary (in thousands of dollars) for elementary school teachers in the United States from 1990 to 2002 is approximated by the model S 1.05t 31.0, 0 β€ t β€ 12 S t where 1990. represents the year, with (Source: National Education Association) corresponding... |
determine the length of time required to perform a particular task in a manufacturing process. The times required by approximately two-thirds of the workers in the study satisfied the inequality t 15.6 1.9 < 1 t is time in minutes. Determine the interval on the where real number line in which these times lie. 108. Hei... |
ize and use algebraic techniques that are common in calculus. Why you should learn it An efficient command of algebra is critical in mastering this course and in the study of calculus. Algebraic Errors to Avoid This section contains five lists of common algebraic errors: errors involving parentheses, errors involving f... |
Exponents have priority over coefficients. Do not move term-by-term from denominator to numerator. Potential Error 5x 5x Correct Form 5x 5x Leave as Leave as x 2 a 2. x a. 1 bx Errors Involving Dividing Out Potential Error a bx a a ax a 1 x 2x a bx a a ax a 1 x 2x Correct Form bx a a a a1 x a 1 1 2 3 2 Comment Radical... |
Useful Calculus Form x12 2x 32 x12 1 x x 2 1 2x x 2 2x Comment Divide each term by x12. Rewrite the fraction as the sum of fractions. Add and subtract the same term. x 2 1 x 1 x 2 1 2x 2 2 x 2 2x 1 2x 2 x 2 2x 1 2 x 1 2 Rewrite the fraction as the difference of fractions Use long division. (See Section 2.3.) Use the m... |
2 9 4 4x 2 9 Solution To write the expression on the left side of the equation in the form given on the right side, multiply the numerators and denominators of both terms by 1 4. 4y2 4x2 91 4 1 4 4y21 4 1 4 x2 9 4 y2 1 4 4x2 9 Now try Exercise 29. Example 5 Rewriting with Negative Exponents Rewrite each expression usi... |
5 x 6x y 6x y 1 x y1 x2x 1 2 2x 2 x 2 1 2y 12y x 1 In Exercises 19β38, insert the required factor in the parentheses. 7x2 10 3 4x 1 7 10 1 4 2 19. 21. 23. 24. 25. 26. 27. 28. 29. 30. 2 20. 1 5 3x 5 1 3x 2 5 3x 2 1 x2x3 14 x3 143x2 x1 2x 23 1 2x234x 22. 3 1 x2 3x 73 1 x 2 2x 32 3 x 6x 5 3x3 2 4x 6 x 2 3x 7 3 x 1 x 2 2x... |
xd 12/6/05 2:22 PM Page A76 A76 Appendix A Review of Fundamental Concepts of Algebra 52. 53. 54. 55. 56. 57. 58. 59. 60. 4x2 9128x 4x 2 9122 2x 31 4x 2 912 2 x 2 34x 323 x 313x 214 x 234 2 2 2x 112 x 22x 112 23x 113 2x 1 1 3 3x 1 23 3x 1233 x 1 1 2 2x 3x 2122 6x 2x 3x 212 x 1 2 2 1 x2 412 1 x2 4122x 2 2x 1 1 x2 6 x2 51... |
that it is correct. Start 2 mi Swim x 4 β x Run 2 mi Finish 67. x n x3n x3n2 69. x 2n y2n x n y n2 68. 70. x n2n x 2nn 2x 2n2 x 2n x 3n x 3n x2 x 5n x 3n x2 71. Think About It You are taking a course in calculus, and for one of the homework problems you obtain the following answer. 2x 152 1 6 2x 132 1 10 (a) Find the ... |
3 ) β4 β6 (4, β 5) (b) (c) 210 2, 3 (5, 4 82 3 1, 7 6 (b) (c) C H A P T E R 1 5000 4500 4000 3500 3000 10 11 12 13 Year (6 β 1996) 25. 5 23. 8 27. (a) 4, 3, 5 29. (a) (b) 10, 3, 109 42 32 52 102 32 1092 (b) (b) 10 (c) 5, 4 (9, 7) 31. (a) y 12 10 8 6 4 2 (1, 1) β 2 2 4 6 8 10 x β 5 2 β2 β 3 2 β1 β 1 2 1 2 39. (a) y 8 6... |
β 1996) x (c) Answers will vary. Sample answer: Technology now enables us to transport information in many ways other than by mail. The Internet is one example. 71. y 8 6 4 2 (β3, 5) (β2, 1) (3, 5) (2, 1 (β7, β3) β 4 β 6 β 8 (7, β3) (a) The point is reflected through the y-axis. (b) The point is reflected through the ... |
β2 β1 1 2 3 4 x β1 β2 33. 37. y 7 6 5 4 2 1 (0, 3) 391 3( β3, 0 ( β4 β 3 β 2 41. y 43. (3, 0 10 12 10 8 6 4 2 β 2 β 2 (0, 6) (β1, 0) (0, 1) β 2 1 2 3 4 (6, 0) 2 4 6 8 10 12 x (0, β1) β2 β3 45. 10 47. β 10 10 β 10 10 β10 Intercepts: β 10 6, 0, 0, 3 Intercepts: 3, 0, 1, 0, 0, 3 49. 10 51. 10 β10 10 β10 10 Intercept: 53.... |
with respect to the x-axis if, is also on the graph. 79. The viewing window is incorrect. Change the viewing is on the graph whenever x, y x, y window. Answers will vary. 9x5, 4x3, 7 107x x 83. 3t 87. 22x 81. 85. Section 1.3 (page 34) Vocabulary Check (page 34) 17. m 0; y -intercept: y 0, 3 (0, 3) 5 4 2 1 β3 β2 β1 β1 ... |
-intercept. y 2 1 β1 1 2 3 β1 β2 11. m 1 2; y -intercept: 00, 4) β 15. β2 m 7 6; y -intercept: y 0, 5 (0, 51 β2 333200_01_AN.qxd 12/9/05 1:23 PM Page A81 434, 0) x β1 1 2 3 4 Answers to Odd-Numbered Exercises and Tests A81 45. x 6 y 6 4 2 β4 β2 2 4 x (6, β1) β2 β4 β6 632 β3 β4 β5 β6 β7 β8 ), β8 ) 7 3 β1 β2 47 49. y 5x... |
to line (a) and line (b). x 8 (c10 β8 14 (b) (a) 3x 2y 1 0 89. 93. (a) Sales increasing 135 units per year 91. 80x 12y 139 0 (b) No change in sales (c) Sales decreasing 40 units per year 95. (a) Salary increased greatest from 1990 to 1992; Least from 1992 to 1994 (b) Slope of line from 1990 to 2002 is about 2351.83 (c... |
(0 β 1990) (c) Answers will vary. Sample answer: y 11.72x 14.1 (d) Answers will vary. Sample answer: The y-intercept indicates that initially there were million subscribers which doesnβt make sense in the context of this problem. Each year, the number of cellular phone subscribers increases by 11.72 million. 14.1 (e) ... |
) 4 (c) 6 (c 41 43, Β±1 4 3 Β±3 51. 49. 47. 57. All real numbers 45. 5 55. 3, 0 59. All real numbers t except 61. All real numbers y such that 63. All real numbers x such that 65. All real numbers x except x 0 53. 2, 1 333200_01_AN.qxd 12/9/05 1:23 PM Page A83 Answers to Odd-Numbered Exercises and Tests A83 except s 4 10... |
h d2 30002, d β₯ 3000 103. False. The range is 105. The domain is the set of inputs of the function, and the 1,. range is the set of outputs. 107. (a) Yes. The amount you pay in sales tax will increase as the price of the item purchased increases. (b) No. The length of time that you study will not neces- sarily determi... |
. to x2 is 0. to x2 is to 73. Odd; origin symmetry x 3 5 3 11 1 4. 63. The average rate of change from 65. The average rate of change from 67. The average rate of change from 69. The average rate of change from 71. Even; -axis symmetry 75. Neither even nor odd; no symmetry 77. 81. 85. (a) h x 2 4x 3 L 1 L 4 y 2 2y 2 83... |
95. (a) (b) 0 0 s 16t2 120 140 0 0 8 4 (c) The given value is not in the domain of the function. h 4, h 0 115. Section 1.6 (page 71) Vocabulary Check (page 71) 1. g 6. e 2. i 7. f 3. h 8. c 4. a 9. d 5. b 32 (c) Average rate of change (d) The slope of the secant line is negative. 32t 120 (e) Secant line: (f) 140 0 0 4... |
9 8 10 3 4 β4 (d) (d) 3 19 (d) 1 39. 22 (d) 41 y 2 1 β4 β3 β2 β1 1 2 3 4 x 11 (b) 4 (c) 6 (c) (b) 2 (b) 3 (b) y 4 3 2 x 3 4 43 412 β5 β6 y 4 3 1 β1 1 2 3 4 x β1 β2 y 5 4 3 1 β4 β3 β2 β1 1 2 3 4 x 49. β4 β3 β2 β3 y 5 4 3 2 1 β1 β1 β2 β3 1 2 3 4 x 47. y 10 8 4 2 β4 β2 2 4 6 8 β2 x 51. (a) 8 β9 9 β4 (b) Domain: Range:, ; ... |
d) These values are quite close to the actual data values. 69. False. A piecewise-defined function is a function that is defined by two or more equations over a specified domain. That domain may or may not include - and -intercepts 3x 6, 5x 16 75. Neither 71. 73 Section 1.7 (page 79) Vocabulary Check (page 79) 2. f x; ... |
1, 3) (3, β 2) β1 β1 1 (2, 0) x 4 (4, β1) (2, 4) (β 3, 4) (0, 3) (β 1, 3 (β1, 0) β3 β1 1 2 (β3, β1) β 1 x β3 β2 β1 (0, 0) x 2 β1 (2, β1) (f) (5, 1) (3, 0) 1 2 4 5 (β2, 2) x y y 3 2 1 ) 0, ) 3 2 (1, 0) β2 β1 1 β1 β2 x ) 3, β ) 1 2 x (eg) (0, β4) (β1, 4) (2, β 3) y 5 4 3 2 1 (0, 3) 2(, 01 ( β ( 3 (, β1 y x 52 3 y 1 x 12 ... |
2 333200_01_AN.qxd 12/9/05 1:23 PM Page A89 Answers to Odd-Numbered Exercises and Tests A89 (c2 β1 1 x β2 β3 β4 (d) gx 2 f x 5 (c) (d) gx 27. (a) f x x 35. (a) f x x (b) Horizontal shrink of of (c) y 1 3, f x x (d) gx f 3x 29. (a) f x x3 (b) Vertical shift two units upward, and horizontal shift one f x x3 unit to the r... |
of 57. Reflection in the x 2 x2 y 1 y 4x2 3 y 3x 3 y 2 x3 -axis and vertical shrink of y x2 ; 59. Reflection in the -axis and vertical shrink of 23 2 61. 65. (a) 63. y x 3 (b) 67. (a) Horizontal stretch of 0.035 and a vertical shift of 20.6 units upward 30 35 ) 40 ( 20 25 15 t 8 4 12 20 Year (0 β 1980) 16 (b) 0.77-bil... |
x2 6 1 x 55. T 3 4 x 1 15 x2 9. (a) (c) (d) x2 6 1 x x2 61 x x2 61 x 1 x ; all real numbers x such that 11. (a) x 1 x2 (b) x 1 x2 (c) (d) x; all real numbers x except 1 x 3 x 0 17. 9t 2 3t 5 19. 74 13. 3 21. 26 25. 15. 5 23 ( 300 250 200 150 100 50 T B R x 10 20 30 60 Speed (in miles per hour) 40 50 27. y 5 4 3 f 57. ... |
.6t 132 A N4 84.16 A N8 81.44 A N12 123.84 10.20t 92.7 y1 3.357t 2 26.46t 379.5 y2 0.465t2 9.71t 7.4 y3 y2 y1 represents the amount spent on health care in the United States. 720 2.892t 2 6.55t 479.6; y3 this amount y1 + y2 + y3 y2 y1 y3 10 61. (a) (b) (c) 63. (a) (d) In 2008, $1298.708 billion is estimated to be spent... |
x g2x x y 3 2 1 f g x β3 β2 1 2 3 β2 β3 15. (a) f gx fx 1 7 7x 1 7 1 x g f x g7x 1 7x 1 1 7 x (b) y 5 4 3 2 1 g f 17. (a) f gx f 38x 38x3 8 x g f x gx3 8 38x3 8 x x2 4 4 x g f x gx 4 x 42 4 x (b) y 10 8 6 4 2 g f 21. (a) (b) 2 4 6 8 10 x f gx f 9 x, x β€ 9 9 9 x2 x g f x g9 x2, x β₯ 0 9 9 x2 x y 12 9 6 f g β12 β9 β6 β3 ... |
in C H A P T E R 1 39. (a) (b) β20 f 1x 1 f 2 4 6 8 x β2 β 2 f 1 (c) The graph of (d) The domains and ranges of f and is the same as the graph of f. f 1 are all real 47. (a) (b) 0 β€ x β€ 2. numbers x such that f 1x 4 x y 4 3 2 1 β1 f = f β2 β3 f 1 (c) The graph of (d) The domains and ranges of f and is the same as the ... |
load y (c) 0 < x < 92.11 yields the year for a given number of miles traveled β 6 β 4 2 4 6 x 0 600 β 6 (c) The graph of y x. the line f 1 is the reflection of the graph of f in (d) The domains and ranges of f and f 1 are all real 53. (a) (b) numbers. f 1x 5x 4 6 4x 3 β 2 β1 f f 1 2 3 x β1 f (c) The graph of y x. the ... |
64 100 4 6 2 Year (2 β 1992) 8 10 12 The model is a good fit for the actual data. 3. y y 5 2x 3 y 1 4x 3 7. (a) and (b) y 240 220 200 180 160 140 ) 100 80 60 40 20 2 4 6 8 10 x 15. x x y k x2 2 2 4 8 6 8 18 32 10 50 C H A P T E R 1 y 50 40 30 20 10 t 2 4 6 8 10 x 108 12 36 84 60 Year (12 β 1912) y 1.03t 130.27 (c) (e)... |
) Yes. k1 k4 C 4300 d (c) (d) 6 (e) 1433 meters 0 0 73. (a) 0.2 25 0 (b) 0.2857 microwatt per square centimeter 6000 55 75. False. y will increase if k is positive and y will decrease if k is negative. 77. True. The closer the value of 79. The accuracy is questionable when based on such limited is to 1, the better the ... |
5 β 4 β3 β2 x x β1 β1 1 2 37. Center: 0, 0; Radius: 3 y 4 2 1 (0, 0) x β4 β2 β1 β1 1 2 4 β2 β4 39. Center: 2, 0; Radius: 4 y (β2, 0) β8 β4 β2 6 2 β2 β6 x 4 333200_01_AN.qxd 12/9/05 1:23 PM Page A98 A98 Answers to Odd-Numbered Exercises and Tests 41. Center: 2, 1; 1 y Radius: 6 β1 55. y 3 2x 5 57. y 1 2x 0, β 5 10 12 x ... |
.00) β0.75 0.75 m 1 2 m 5 11 β3 3 β1 β0.75 333200_01_AN.qxd 12/9/05 1:23 PM Page A99 95. 4 97. 1 2 2 101. Odd 103. f x 3x 99. Neither even nor odd 1056 β4 107. 109. y 6 4 β2 β4 β 111 x3 115. 117. (a) f x x2 β1 1 2 3 4 5 6 x 113. y 6 3 β12 β9 β6 β3 3 6 9 12 15 x β12 β15 (b) Vertical shift of nine units downward (c β 10 ... |
real numbers. f 1x x2 11 f β1 β1 2 3 4 5 x (c) The graph of y x. the line f 1 is the reflection of the graph of f in (d) f has a domain of has a domain of 1, 0, and a range of and a range of 0, ; 1,. f 1 151. x β₯ 4; f 1x x 2 4 153. (a ( 65 60 55 50 45 t 5 6 7 8 9 10 11 12 Year (5 β 1995) (b) The model is a good fit fo... |
9. (a) x 12 y 32 16 17x 10y 59 0 4x 7y 44 0 1 8 10 β€ x β€ 10 0, Β±0.4314 1 28 (b) (c) 10. (a) 11. 12. (a) (b) 0.1 β1 1 y 163. True. If x 1ky. is directly proportional to x to a set A Therefore, 165. A function from a set A in the set to each element set B. B x x, is directly proportional to y kx, y. then is a relation t... |
5 3x 4 24x3 18x2 120x 68 9x 4 30x2 16 1 2x32 x 2x x x 2x 1 2x32, x > 0 2x, x > 0 x 1 2x32 d) (b) (f), x > 0 20. 22. 24. f 1x 3x 8 f 1x 1 A 25 6 3 x23, x β₯ 0 25. b 48 a xy 21. No inverse 23. v 6s Problem Solving (page 125) (c) 1. (a) (c) W1 5,000 2000 0.07S (b) W2 2300 0.05S (15,000, 3,050) 0 0 30,000 Both jobs pay the... |
β3 y 3 2 β1 β2 β3 1 2 3 1 2 3 x x 333200_02_AN.qxd 12/9/05 1:27 PM Page A103 Answers to Odd-Numbered Exercises and Tests A103 (e) y (f) y (c3 β2 β1 3 1 β1 β2 β3 1 2 3 x 13. Proof 15. (a) x f f1x (b 1x 5 1 3 5 (c) x 3 2 f f 1x 4 0 (d) x f 1x Chapter 2 Section 2.1 (page 134) Vocabulary Check (page 134) 1. nonnegative in... |
s: x 5 5 Β± 6, 0 21. y 23. 20 16 12 8 4 β4 4 8 12 16 4, 0 Vertex: Axis of symmetry: 4, 0 x-intercept, 1 1 Vertex: Axis of symmetry: No x-intercept 2 β 4 1, 6 Vertex: Axis of symmetry: x-intercepts: x 1 1 Β± 6, 0 25. y 27. y 4 β 8 4 8 16 x x x β8 4 2, 3 Vertex: Axis of symmetry: x-intercepts: x 2 2 Β± 6, 0 y x 12 4 43. f x... |
1600 x 25 feet, y 33 1 3 feet (c) 2000 0 0 60 x 25 feet, y 33 1 3 feet x 252 5000 A 8 (d) 3 3 79. 20 fixtures 77. 16 feet 83. (a) $14,000,000; $14,375,000; $13,500,000 81. 350,000 units (e) They are identical. (b) 24; $14,400 85. (a) 5000 (b) 4299; answers will vary. (c) 8879; 24 0 0 43 333200_02_AN.qxd 12/9/05 1:27 P... |
. a 9. (a) 2. g 6. e y 3. h 7. d 4. f 8. b (b) β 3 β 2 (cd) β3 β 11. (a) (b) y 6 5 4 3 2 1 β30 29. (a) 3 1 2 4 5 x (b) even multiplicity; number of turning points: 1 (c) 4 β18 18 31. (a) 2, 1 β20 (b) odd multiplicity; number of turning points: 1 (c) 4 3 2 1 β2 β3 β4 β5 y 4 3 2 1 β4 β3 β2 2 3 4 x β4 β4 333200_02_AN.qxd ... |
(b) (d) y 12 4 β4 β8 (β3, 0) β12 β8 β4 (0, 0) (3, 0) 4 8 12 x 69. (a) Rises to the left, rises to the right (b) No zeros (d) y (c) Answers will vary. β4 4 41. (a) Β±2, 3 β5 (b) odd multiplicity; number of turning points: 2 (c) 4 8 6 2 β8 β16 43. (a) 12 β2 β4 7 6 x -intercepts: 2, 0 (b) (d) The answers in part (c) match... |
3; 0.879, 1.347, 2.532 87. 89. (a) 2, 1, 0, 1; 1.585, 0.779 V l w h 36 2x36 2xx x36 2x2 (b) Domain: (c) 0 < x < 18 x V 1 2 3 4 5 6 7 1156 2048 2700 3136 3380 3456 3388 24 inches 24 inches 6 inches (d) 3600 0 0 18 x 6 A 2x2 12x 0 inches < x < 6 (b) inches 91. (a) (c) (d) V 384x2 2304x x 3, the volume is maximum at When... |
3 β2 β1 1 2 x β2 Section 2.3 (page 159) Vocabulary Check (page 159) 1. dividend; divisor; quotient; remainder 2. improper; proper 4. factor 5. remainder 3. synthetic division 1. Answers will vary. 3. 6 5. 2x 4 β9 9 7. 13. 17. 21. 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 7 11 11. x 2 x2 2x 4 2x 11 x2 2x 3 3x2 2x 5 19. 15... |
, 5, 4 20 β6 6 β180 61. (a) Answers will vary. x 7 f x x 72x 13x 2 7, 1 (b) 2, 2 3 (c) (d) (e) (c) (e) 320 β9 3 14 β6 6 β6 β40 63. (a) Answers will vary. x 5 f x x 5x 52x 1 (b) (d) Β± 5, 1 2 65. (a) Zeros are 2 and x 2 (b) Β±2.236. f x x 2x 5x 5 3.732. (c) 2, 0.268, and 67. (a) Zeros are (b) (c) 2x2 x 1, x 3 2 x 2 h t t ... |
i a 10, b 6 2 33 i 1 6i 3 32 i 5 i 10 25i, 8 10 41i 41 120 27 1681 23 21 52 75 310i 23. 12 30i 6 3i, 45 8, 8 43. 3 4 5i 5 1 2 29. 37. 20 55. 10 51. 5 2i 1681i 61. 49. 31. 24 39. 19. 4 7 6i 9 40i 1 6 33. 1 5 i, 6 5i 45. 5 6i 57. 62 949 297 949i 65. 1 Β± i 2 Β± i 1 2 Β± 515 7 5 7 5i 69. 5 2, 3 2 75. 1 6i 71. 2 Β± 2i 77. 83. ... |
, Β±3 2, Β± 15 2, Β± 45 2 17. 1 2, 1 (c) 2, 1, 2 (c) 1 4, 11, Β±3, Β± 1 2, Β± 3 2, Β± 1 4 10 x β 4 β 6 Β±1, Β±2, Β±4, Β±8, Β± 1 2 16 β8 β4 8 27. (a) (b) 29. (a) (b) 31. (a) Β±1, Β±1.414 f x x 1x 1x 2x 2 0, 3, 4, Β±1.414 hx xx 3x 4x 2x 2 39. (b) 35. (a) (b) x3 x2 25x 25 3x4 17x3 25x2 23x 22 x3 4x 2 31x 174 37. 41. 43. (a) (b) (c) 45. ... |
1 4 Β±2, Β± 3 2 102. c x 2x 1 2 ix 1 2 i 5x 1x 1 5 ix 1 5 i (c) 1 2, 1, 2, 4 (b) (c) 9 x x 9 β 2x x 5 β 2 1 x V x9 2x15 2x Domain: 0 < x < 9 2 V 125 100 75 50 25 1, Β±3, Β± 1 2, Β± 3 2, Β± 1 4, Β± 3 4, Β± 1 8, Β± 3 (b) 6 8, Β± 1 (c) 16, Β± 3 1, 3 16, Β± 1 4, 1 8 32, Β± 3 32 x 5 4 2 3 1 Length of sides of squares removed β1 3 β2 10... |
4) 4 3 2 1 β 1 β 2 137. (2, 2) (4, 2) (0, 0) 2 3 4 5 6 x (2, 4) y y 4 3 (0, 2) (1, 2) (β1, 0) β 2 β 1 xx 1 2 Section 2.6 (page 193) Vocabulary Check (page 193) 1. rational functions 3. horizontal asymptote 2. vertical asymptote 4. slant asymptote x 0.5 0.9 f x 2 10 0.99 100 0.999 1000 x 1.5 1.1 f x 2 10 x 5 10 f x 0.2... |
(a) Domain: all real numbers x except 0, 1 (b) y-intercept: (c) Vertical asymptote: x 2 Horizontal asymptote 333200_02_AN.qxd 12/9/05 1:27 PM Page A112 A112 (d) Answers to Odd-Numbered Exercises and Tests y 2 1 β1 β2 ( 0, ( 1 2 x β 3 β1 29. (a) Domain: all real numbers x except 0, 1 (b) y-intercept: 2 (c) Vertical asy... |
: x 2 x 2, 3 Horizontal asymptote: y y 1 (d) 6 4 2 β6 β4 β2 (0, 0) β4 β6 x 4 6 333200_02_AN.qxd 12/9/05 1:27 PM Page A113 43. (a) Domain: all real numbers x except 2, 0 1 0, 1 (b) x-intercept: y-intercept: 3 (c) Vertical asymptote: Horizontal asymptote: x 3 2 y 1 x 3 2, 2 (d 45. (a) Domain: all real numbers t except 1,... |
β4 2 β2 β4 β6 53. (a) Domain: all real numbers x except b) No intercepts (c) Vertical asymptote: Slant asymptote: x 0 y 2x (d) y 6 4 2 y = 2x β6 β4 β2 2 4 6 x β6 55. (a) Domain: all real numbers x except x 0 (b) No intercepts (c) Vertical asymptote: x 0 Slant asymptote: y x 333200_02_AN.qxd 12/9/05 1:27 PM Page A114 A... |
170 million; $765 million p 100. (c) No. The function is undefined at (b) 1500 deer 75. (a) 333 deer, 500 deer, 800 deer 77. (a) Answers will vary. (b) 4, (c) 200 63. (a) Domain: all real numbers x except x 1, 2 (b) y-intercept: x-intercepts: 0, 0.5 0.5, 0, 1, 0 4 0 40 11.75 inches 5.87 inches 333200_02_AN.qxd 12/9/05 ... |
76 β4 β8 β2, 5 1, 0 2 4 11. 13. β6 β5 β4 3, 2 15. β3 β2 β1 0 1 2 β3 β2 β 25. 27. (, 0 0, 3 2 1 2 29. 33. 0 β1 β2 2, 0 2, 1 6 β2 β5 (a) (b) x β€ 1, x β₯ 3 0 β€ x β€ 2 37., 1 0, 1 x 2 31. 2, 35. 8 β12 12 β8 2 β€ x β€ 0, (a) (b) 392 β1 0 1 2 β2 β1 0 1 2 3 4 5 41. 5, 15 5 3 6 9 12 15 18 45. 3 4, 3 6, β 3 4 3 x x 43. 5, 3 2 1, β... |
, a > 0 and 83. (a) If c β€ 0, b (f) Answers will vary., 3, 3, 1, 1, 4, and 81., 230 230, a > 0 can be any real number. If and c > 0, b < 2ac or b > 2ac. (b) 0 2x 52 85. 87. x 3x 2x 2 89. 2x2 x Review Exercises (page 208) 1. (a Vertical stretch (b4 β3 β2 β1 1 2 3 4 x x β3 β4 Vertical stretch and reflection in the x-axis... |
to the left, falls to the right 31. Rises to the left, rises to the right 33. 35. 37. 0, even multiplicity; 39. (a) Rises to the left, falls to the right 7, 3 2, 0, Β± 3, odd multiplicity; turning point: 1 odd multiplicity; turning points: 2 5 3, (b) 1 odd multiplicity; turning points: 2 (c) Answers will vary. (d) y 4 ... |
2, Β± 15 2, Β± 1 2, Β± 5 89. 4, 3 1, 3, 6 95. 91. 93. 3x4 14x3 17x2 42x 24 97. 4, Β±i 99. 0, 1, 5; f (x x x 1x 5 103. 4, 2 Β± 3i; gx x 42x 2 3i 105. 107. Two or no positive zeros, one negative zero 109. Answers will vary. 111. Domain: all real numbers x except 113. Domain: all real numbers x except 115. Vertical asymptote:... |
2 x 1 2 127. (a) Domain: all real numbers x (c) Horizontal asymptote: y 6 (b) Intercept: 0, 0 3 2 1 (0, 0) β3 β 2 β1 1 2 3 β2 β3 x 133. (a) Domain: all real numbers x except (b) y-intercept: x-intercepts: (c) Vertical asymptote: Slant asymptote: y (d) 0, 0.5 2 3, 0, 1( ( 0, β 1 2 ( β2 β1 β2 (1, 0) 2 3 4 x 135. $0.50 i... |
6 9 x 2 8. (a) 10. 11. 12. 14. 3 5i (b) 7 9. 2 i 13. f x x4 9x3 28x2 30x f x x4 6x3 16x2 24x 16 2, Β± 5i 2, 4, 1 Β± 2 i x -intercepts: No -intercept Vertical asymptote: Horizontal asymptote: y 2, 0, 2 (β2, 0) (2, 0) β2 β1 1 2 β2 x 15. 1.5, 0 x -intercept: 0, 0.75 y -intercept: Vertical asymptote: Horizontal asymptote2 β... |
1 r n nt 1. 946.852 7. d 11. 8. c x 2 1 3. 0.006 9. a 5. 1767.767 10. b 0 1 1 2 0.5 0.25 2 β 1 1 2 3 β 1 13. x 2 1 f x 36 6 0 1 x 1 2 0.167 0.028 15. x f x 2 1 0 0.125 0.25 0. 17. Shift the graph of f four units to the right. 19. Shift the graph of f five units upward. 21. Reflect the graph of f in the x-axis and y-axi... |
. 43. b 44. a 61. 2.913 ln 4 x ln 1.6487... 1 2 59. 2 3 Domain: x-intercept: Vertical asymptote: 1, 2, 0 x 1 3 2 1 β1 1 2 3 4 5 x β1 β2 β3 y 2 1, 0 Domain: 1, 0 x-intercept: Vertical asymptote: x 0 33. y 6 4 2 β2 β 4 β 6 35. β3 β2 β1 x 1 β2 x 0 73. 2 75. 3 0, 9, 0 Domain: x-intercept: Vertical asymptote: 2 4 6 8 10 12 ... |
53. 57. 19. 27. 2.4 33. 29. 9 35. 7 41. 47. 4 log8 x ln x ln y 2 ln z 1 51. 2 log2 4 ln x 1 1 2 log4 5 log4 x 1 2 ln z ln z 2 lnz 1 1 3 ln x 1 2 log5 x 2 log5 y 3 log5 z 3 ln x 1 4 4 lnx2 3 3 ln y 55. 61. 59. 65. log2 x 4 2 67. log3 45x 4 ln 77. 73. x x2 44 xz3 y2 3y y 42 y 1 log2 32 log2 4; 60 dB log8 log2 32 10log I... |
1 5 y 2 1 β1 β 333200_03_AN.qxd 12/9/05 1:30 PM Page A124 A124 Answers to Odd-Numbered Exercises and Tests 103. 107. 3x4 2y 3, x 0 1 1, 3 109. 105. 1, x 0, y 0 1 Β± 97 6 Section 3.4 (page 253) Vocabulary Check (page 253) 1. solve x y 2. (a) 3. extraneous (b) x y (c) x (d) x (b) No (b) Yes 1. (a) Yes 3. (a) No 5. (a) Ye... |
population of 2430 thousand y 5e0.4024x 33. 2003: population of 2408.95 thousand (c) 2018 k 0.2988; 5,309,734 hits (b) 12,180 years old V 6394t 30,788 37. 41. (a) 43. (a) (c) 32,000 39. 3.15 hours 4797 years old (b) V 30,788e0.268t The exponential model depreciates faster. 0 0 t (d) 4 1 3 V 6394t 30,788 24,394 11,606 ... |
0 49. (a) 203 animals 1200 (c) (b) 13 years Horizontal asymptotes: y 0, y 1000. The population size will approach 1000 as time increases. (b) 108.3 199,526,231 0 0 40 51. (a) (c) 107.9 79,432,823 104.2 15,849 53. (a) 20 decibels (c) 40 decibels (b) 70 decibels (d) 120 decibels 55. 95% 105.1 61. 65. (a) 150,000 57. 4.6... |
56.529 10. b 3. 0.337 1. 76.699 9. a 7. c 11. Shift the graph of f one unit to the right. 13. Reflect f in the x-axis and shift two units to the left. 15.25 4.063 4.016 17. x 2 f x 0.377.65 7.023 18.61 β 3 β 6 β 9 β 12 β 15 333200_03_AN.qxd 12/9/05 1:30 PM Page A127 19. x f x 1 0 1 4.008 4.04 4. 21. x f x 2 1 3.25 3. 2... |
0.632 y 1 2 1.718 1 6.389 0 0 x 6. 7. 10 β 4 β3 β 2 β1 1 2 3 4 β. (a) 9. 0.89 (b) 9.2 1 2 5.699 1 6 3 2 6.176 2 4 6.301 6.602 Vertical asymptote 333200_03_AN.qxd 12/9/05 1:30 PM Page A129 10. 5 0 x f x y 7 9 11 13 1.099 1.609 1.946 2.197 Vertical asymptote: x 4 4 2 β2 β4 2 6 8 x 11.099 2.609 2.792 2.946 Answers to Odd... |
except x 1 4 (b) x 1 x2. (a) ; (d) (b) 5x 2 x 3 4x 1 x 1 x2 1 x2x 1 x 1 x 1 x2 1 2x 12 Domain of Domain of x 2 Domain of f g: g f: (b) (d) ; 10. (a) (c) 11. (a) 12. (a) h1(x) 1 5 x 82 5 13. Yes; 15. 16. y 3 4 β8 β6 β2 2 4 6 x Domain: all real numbers x such that x β₯ 1 2x2 6 (b) all real numbers x such that all real nu... |
, 0) 2 4 6 8 28. x β€ 2 or 0 β€ x β€ 2 β3 β2 β1 0 1 2 3 x x 29. All real numbers such that x x < 5 or x > 1 β6 β5 β4 β3 β2 β1 0 1 2 x 30. Reflect f in the x-axis and y-axis, and shift three units to the right. β10 7 f β7 11 g 31. Reflect f in the x-axis, and shift four units upward. 6 f g 8 β6 33. 0.067 34. 1.717 35. 0.28... |
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