Transcription
Math 1314College AlgebraProblems andAnswersFall 2010LSC – North Harris
Material taken from:Weltman, Perez, Tiballiunpublished material“College Algebra” versionby Stitz and Zeager37Go to LSC‐North Harris Math Departmentwebsite for updated and correctedversions of this material.Math Dept Website: nhmath.lonestar.edu
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Section 2.4-Absolute Value1. x 8 or x 83. x 6 or x 65. x 7. x 4 or x 39. x 10 or x 211. x15. No solution17. x 83819. x 8 or x 34225. x or x 153131. x 313. x 2 or x 21. x 5223. x27. x 5 or x 429. x33. No solution35. x37. x 3 or x 139. x 2 or x 441. x51or x 3545. x 3 or x 43. x 51.( 5, 5)-6 -5 -4 -3 -2 -1-6 -5 -4 -3 -2 -1-6 -5 -4 -3 -2 -11 2 3 4 x( ,11 [1, )67. No Solution-6-5-4-3-2-1 1 2 3 4 5 6 7 8 910 x61.2 x( 5, 4 )-6 -5 -4 -3 -2 -1x1 2 3 4 5 657. [ 5, 9]-12 -10 -8 -6 -4 -263.( , 1) (1, )x55. [ 4, 2]59.11753.1 2 3 4 5 655or x 447 or x 221014 or x 33148 or x 5510 2 or x 317 13 or x 121211 2-1( , 5 [ 7, )1 2 3 4 5 6 7 8 9 101165. x 1 2 3 4 5 6-1x13-23 11 69. 3, 3 Page 15x-132137383231 x3 10 11 4 x3 3
1 71. , (3, )3 -275.-1173. x 24 x3( , )-2 -3 -1 -122121 2 3 4 5 6 7 8 9 1011 -287.2/31 2 3 4 5 6 7 8x-389.x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1-3-2-193.1-2-12 x-3Page 16123 x( 1, 5)1 2 3 4 5 6 7 x-2 -110 2 91. , , 3 3 -4x12 1339 7 85. , , 4 4 ( 9, 1)-54 x2x( , 1 [5, )-3 -2 -12 x -3 -8 -7 -2 -5 -4 -1 -2 -13 3 33 3383.322 7 81. ,1 3 ( 4,8)1( , 2 3 , 77.-479.11 1 or , , 22 2 ( , 0 ) (1, )-2-1123 x
Absolute Value InequalitiesSolve the inequalities. Graph your answer on a number line. Write answersin interval notation.1.x 318.3 x 1 2 42.x 519.9 2 x 2 13.x 2 720.2 x 4 5 84.3x 5 1021.2x 3 125.x 2 4322.2x 5 923.2 x 3 5 1324.4 x 1 7 1325.2x 1 12 526.4 5 x 427. 2x 5 66.2x 3 107.4 x 2 3 98.3 x 1 7 109.1 4 x 7 210.1 2x 4 111.x 428.x 1 0212.x 729.6 7x 013.x 1 830.x 2 6 314.2x 1 731.6 8 x 415.x 1 3532.x 1 016.3x 4 833.3 x 1 5 517.2 x 10 1634.47 x 9 5Page 17
Absolute Value Inequalities- Answers1.[ 3,3]17.( , 18] [ 2, )2.( 5,5 )18.( , 1) (3, )3.[ 9,5]19.( ,4 ] [5, )4. 5 3 ,5 21 5 20. , , 2 2 5.( 10,14 )6. 13 7 2 , 2 9 15 21. , 2 2 7.22.( , 7] [2, )( 5,1)23.[ 1,7]8.[0,2]24.( , 4 ) ( 6, )9.3 1, 2 25. 10.26.x 5[ 1,2]27.( , )11.( , 4 ] [4, )28.( , )12.( , 7 ) (7, )13.( , 9] [7, )14.( , 3] [4, )15.( , 14 ) (16, )16.( , 4 ] 4 , 3 29. x 30.67( , )31. 32.( , 1) ( 1, )33. 34. Page 18
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Section 2.5—Quadratic Equations 1 1. ,5 3 1 3 3. , 2 2 1 5. , 0 4 7. 2, 2 9. 1, 4 11. 4,5 3 13. 2 15. 7,3 17. 13 3 5 19. 5 11 21. 2 23.25. 3 7 2,1 3 6 27. 2 29. 2 2 1 31. i 3 7 57. 0, 2 3 i 59. 2 33. 2 2 35. 6, 2 37. 3 10 63.39. 5 3i 4 65. , 0 3 5 13 41. 2 2 5 61. 3 4 2 1 3 67. 2 2 3 43. 2 1 5i 69. 3 5 57 45. 4 3 7 71. 7 1 i 59 47. 6 1 2i 73. 3 49. 1 351. 4 i 5 53. 2 55. 3 i 5 Page 34 1 75. , 3 2 1 i 77. 2
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Section 2.7—Miscellaneous Equations 2 1. 0, ,3 5 3. {0, 5, 2} 3 5. ,1, 1 2 2 2 7. 2, , 3 3 1 9. , 2i 3 11. 1 i 3 3,1, 22 13. 1 2, ,1 i 3 2 5 15. 2 17. 19.{4}21.{9}23.{ 1}25.{6}27.{ 1}29.{32}31.{3}33.{ 1}35.{3} 1 37. , 2 3 3 39. , i 2 41.{ 2 13} 7 3 43. 2 1 45. ,8 27 27 1 47. , 8 8 1 49. , 4 4 51.{9} 2 53. , 3 3 1 55. , 2 4 9 15 57. , 4 4 Page 46 1 3 59. , 3 5 61.{ 7,3} 1 63. , 2 2 65.{ 1,15}67.{ 16,11}69.{ 31,33} 80 71. 3
Quadratic Types of EquationsFind all solutions of the equation.10x 2 x 1 24 01.x 4 13x 2 40 014.2.x 4 5x 2 4 015. 3x 2 7x 1 6 03.x 6 2x 3 3 016. 2x 2 7 x 1 4 04.x 6 7x 3 8 017. 7x 2 19x 1 65.3x18. 5x 2 43x 1 18x19. 6x 26. x 1 220. 9x 4 35x 2 4 07.8.9.xx41110. x3 2x2 4x21211. 2x12. 3x13. 4 x224 2x 5x34x1221 1 0 4 0414 5x11 5 6 041133 9x 3x3214 2 0 2 033 5x3 65x32 3 0 2 03 16 021.( x 2)22.( 2 x 5 ) ( 2x 5 ) 6 02(23. 3x 424.(2 x )(Page 47)225. 2 x 126. 7 ( x 2 ) 12 02)12 6 ( 3x 4 ) 9 0 ( 2 x ) 20 02 5 ( x 1) 31( x 2 ) 2 11 ( x 2 ) 4 18
AnswersQuadratic Types of Equations1.x 2 2, 52.x 2, 13.x 3 3, 14.x 2,15.x 125,1276.x 2 2, 3 37.x 18.x 169.x 1,1625610. x 16,115.3 1x ,2 316.x 2,17.x 18.5 1x ,2 919.x 2, 20.x 3i, 21.x 5, 6147 1, 2 332127211.1x ,27822.x 1, 12.x 1, 82723.x 1x , 64824.x 7, 213.25.x 2, 26.x 6563,1814.2 5x ,3 8Page 481332
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Section 2.8-Polynomial and Rational Inequalities 1 1. 2, 3 -2 -5 -4 -1 -2 -13 33 31 2 13 3-19 5. , ( 0, )2 13.-21 121212 , 4 2 4 2-11 2 45233, 2 32, 15.19.1-12234-21227.1322523 , 1 2 3 4 5 , 5 ( 3, 2)-111 2 3 4234 , 5 3, 3 -7 -6 -5 -4 -3 -2 -129. 3, -3623. 1, 1 3 25. 0, , 2 2 -45–4 2 ,1 ( 2, 3) (3, )-142 3-7 -6 -5 -4 -3 -2 -1-17 2– 317. No Solution21.61-5 -4 -3 -2 -1–4 – 23 1, 5 11.-12-117.-5 -9 -4 -7 -3 -5 -2 -3 -1 -1222229. x , 4 (5, )3.1 2 3 431. 2, 5 -2-1121Page 6223456
33. , 1 ( 6, )-2 -135.1 2 3 4 5 6 7-4 -7 -3 -5 -2 -3 -1 -1222237. 6, 5 39.-7 -6 -5 -4 -3 -2 -141.45.1 2 3 4 5 6-243.-1134547.2 4 6-4-2-6-5123-4456 , 6 ( 1, 4)-8-7-6-5-4-3-2-1123456 13 51. , 4,1 2 , 4 ( 2, )-6-7 , 2 ( 4, )-1 , 10 ( 4, )-10 -8 -6 -4 -249.21 12 , 6 -8 2, 4 -3 1 , 2 , 3 2-8 -7 -6 -5 -4 -3 -2 -1Page 631 2
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Section 3.1-Relations and Functions1.65C 432D1-4 -3 -2 -1-1-2A1 2 3 4B3. d 10, M (3, 1)35. d 17, M ( ,1)27. d 2 29, M (3,3) 5 1 9. d 7 2, M , 2 2 109 11 1 ,M , 12 8 4 13. d 23.05 4.8, M ( 0.15, 2)11. d 5 15. d 3 13, M , 2 2 2x h , y1 17. d h, M 1 2 19. B (10,11)1121. ( , 6)223. P 10 6 5, A 2027. (7, 2);( 9, 2)29. ( 3,12);( 3, 2)31. (0,9);(0, 7)37. Center ( 3, 0); radius 37Page 87
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In exercises 25-42 an equation and its graph are given. Find the intercepts of the graph, anddetermine whether the graph is symmetric with respect to the x-axis, y-axis, and/or the origin.25.26.27.654321-4 -3 -2 -1-1-2-36543211 2 3 428.2y x-4 -3 -2 -1-1-2-31 2 3 4429.30.2y 2xx y 231.32.22x y 4y 3 – x y xy –3x33.224x 9y 36Page 99y 1x
34.35.36.y 4–xx y 31x 1237.38.239.222y –x 1x –y 140.41.2y 4–x2x y –442.2y x 22x 4–yPage 100
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In exercises 69-72, sketch a graph that is symmetric to the given graph with respect to the y-axis.69.70.71.72.In exercises 73-76, sketch a graph that is symmetric to the given graph with respect to the x-axis.73.74.Page 102
75.76.In exercises 77-80, sketch a graph that is symmetric to the given graph with respect to the origin.77.78.79.80.Write Algebra81. Explain what it means for the graph of an equation to be symmetric with respect to they-axis, x-axis, or the origin.82. How do you find the intercepts of the graph of an equation?83. Describe a strategy for finding the graph of an equation.Page 103
Section 3.2-Graphs of Equations1.3.4321-4 -3 -2 -1-1-2-3-45.43211 2 3 47.-4 -3 -2 -1-1-2-3-43-4 -3 -2 -1-31 2 3 41 2 3 4-6-9-129.11.15.17.4321-4 -3 -2 -1-1-2-3-41 2 3 413.4321-2 -1-1-2-3-443211 2 3 4 5 6-4 -3 -2 -1-1-2-3-443211 2 3 4Page 104-6 -5 -4 -3 -2 -1-1-2-3-41 2
19.21.23.654321654321-4 -3 -2 -1-1 1 2 3 4 5 64321-6 -5 -4 -3 -2 -1-1 1 2 3 4-2-237. ( 1, 0); symmetric with respect to y-axis, x-axis,and origin39. (0, 2), ( 4, 0); symmetric with respect to x-axisto y-axis29. (0, 0); symmetric with respect to origin31. (0, 2), ( 2, 0), (2, 0); symmetric with respect toy-axis, x-axis, and origin41. (0, 2); symmetric with respect to y-axis45.47.65432121-4 -3 -2 -1-11 2 3 433. No intercepts; symmetric with respect to origin35. (3, 0); symmetric with respect to x-axis(0, 0); symmetric with respect to y-axis27. (0,3), ( 3, 0), (3, 0); symmetric with respect25.43.-4 -3 -2 -1-1-2-3-41 2 3 4-2-3-449.-4 -3 -2 -1-11 2 3 451.4321-2 -1-1-2-3-453.43211 2 3 4 5 6-2 -1-1-2-3-41 2 3 4 5 687654321-2 -1-1-2Page 1051 2 3 4 5 6
55.57.87654321-2 -1-1-259.43211 2 3 4 5 661.-2 -1-1-2-3-443211 2 3 4 5 663.4321-2-1-1 1 2 3 4 5 6 7 8 910-2-3-465.4321-2 -1-1 1 2 3 4 5 6 7 8-2-3-467.-4 -3 -2 -1-1-2-3-443211 2 3 469.-4 -3 -2 -1-1-2-3-471.654321-4 -3 -2 -1-1-273.1 2 3 475.77.79.Page 1061 2 3 4
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Section 3.4-Relations and Functions1.3.5.7.9.Yes, it is a functionNo, not a functionYes, it is a functionYes, it is a functionf ( 3) 5f ( 0) 4f ( 2) 10f ( x 1) 3x 1f ( x ) f ( 2) 3x 2f ( 3 h) f ( 3) 3hf ( x h) f ( x ) 3h11.f ( 3) 1315.f ( 3) 15f ( 0) 9f ( 2) 5f ( x 1) 3x 2 5x 7f ( x ) f ( 2) 3x 7x 2f ( 3 h) f ( 3) 3h 17hf ( x h) f ( x ) 6x 3h 1h17.53f ( 0) undefinedf ( 3) f ( 0) 5f ( 2) 3f ( 2) f ( x 1) 2x 2 4x 3f ( x 1) f ( x ) f ( 2) 2x 4x 2f ( 3 h) f ( 3) 2h 12hf ( x h) f ( x ) 4x 2hh13.f ( 3) 10525x 1f ( x ) f ( 2)5 x 22xf ( 3 h ) f ( 3)5 h3( 3 h)f ( x h) f ( x ) 5 hx ( x h)19.f ( 0 ) 7f ( 2) 5f ( 3) f ( x 1) x 2 2x 10f ( 0) f ( x ) f ( 2) x 6x 2f ( 3 h) f ( 3) h 2hf ( x h) f ( x ) 2x h 4h1613f ( 2) 11x 4f ( x ) f ( 2)1 x 2x 3f ( 3 h ) f ( 3)1 h6( 6 h)f ( x h) f ( x ) 1 h( x 3)( x h 3)f ( x 1) Page 119
21.f ( 3) 6f ( 0) 02f ( 2) 32x 2f ( x 1) x 324 f (x ) f ( ) 3( x 4)x 28f ( 3 h) f ( 3) (h 1)h8f ( x h) f ( x ) ( x 4)( x h 4)h23.25. 5 27. , 2 29. ( , 2] [ 5, ) , 33. 2, 35. , 37. , 3 (3, 4) ( 4, )39. , 41. , 43. 0, 3 3, 45. 5, 31. , 2 ( 2, ) , Page 120
50Relations and Functions1.5.1ExercisesExercisestakenStitzand ZeagerBookthree steps in1.AdditionalSuppose f is a functionthat takesa realfromnumberx and performsthe followingthe order given: (1) square root; (2) subtract 13; (3) make the quantity the denominator ofa fraction withproblemsnumerator 4.areFindp.an expressionfor f (x) and find its domain.Suggested50: 5-102. Suppose g is a function that takes a real number x and performs the following three steps inthe order given: (1) subtract 13; (2) square root; (3) make the quantity the denominator ofa fraction with numerator 4. Find an expression for g(x) and find its domain.3. Suppose h is a function that takes a real number x and performs the following three steps inthe order given: (1) square root; (2) make the quantity the denominator of a fraction withnumerator 4; (3) subtract 13. Find an expression for h(x) and find its domain.4. Suppose k is a function that takes a real number x and performs the following three steps inthe order given: (1) make the quantity the denominator of a fraction with numerator 4; (2)square root; (3) subtract 13. Find an expression for k(x) and find its domain.5. For f (x) x2 3x 2, find and simplify the following:(a) f (3)(d) f (4x)(g) f (x 4)(b) f ( 1) (c) f 23(e) 4f (x)(h) f (x) 4 (i) f x2(f) f ( x)6. Repeat Exercise 5 above for f (x) 2x37. Let f (x) 3x2 3x 2. Find and simplify the following:2af (a)2 (a) f (2)(d) 2f (a)(g) f(b) f ( 2)(e) f (a 2)(c) f (2a)(f) f (a) f (2)(h)(i) f (a h) x 3 x 5,28. Let f (x) 9 x , 3 x 3 x 5,x 3(a) f ( 4)(c) f (3)(e) f ( 3.001)(b) f ( 3)(d) f (3.001)(f) f (2)Page 121
1.5 Function Notation x2 9. Let f (x) 1 x2 x51ifififx 1 1 x 1 Compute the following function values.x 1(a) f (4)(d) f (0)(b) f ( 3)(e) f ( 1)(c) f (1)(f) f ( 0.999)10. Find the (implied) domain of the function.(a) f (x) x4 13x3 56x2 19tt 8 rQ(r) r 8θb(θ) θ 8ryα(y) 3y 8 A(x) x 7 9 x1g(v) 14 2vw 8 u(w) 5 w(j) s(t) (b) f (x) x2 4x 4(c) f (x) 2x 36 (d) f (x) 6x 26(e) f (x) 6x 2 3(f) f (x) 6x 26 (g) f (x) 4 6x 2 6x 2(h) f (x) 2x 36 36x 2(i) f (x) 2x 36(k)(l)(m)(n)(o)(p)11. The population of Sasquatch in Portage County can be modeled by the function P (t) 150t, where t 0 represents the year 1803. What is the applied domain of P ? What ranget 15“makes sense” for this function? What does P (0) represent? What does P (205) represent?12. Recall that the integers is the set of numbers Z {. . . , 3, 2, 1, 0, 1, 2, 3, . . .}.8 Thegreatest integer of x, bxc, is defined to be the largest integer k with k x.(a) Find b0.785c, b117c, b 2.001c, and bπ 6c(b) Discuss with your classmates how bxc may be described as a piece-wise defined function.HINT: There are infinitely many pieces!(c) Is ba bc bac bbc always true? What if a or b is an integer? Test some values, makea conjecture, and explain your result.8The use of the letter Z for the integers is ostensibly because the German word zahlen means ‘to count.’Page 122
1.5 Function Notation1.5.253Answers41. f (x) x 13Domain: [0, 169) (169, )4x 13Domain: (13, )2. g(x) 5. (a) 2(b) 6(c)6. (a)(b)(c)(d)(e)7. (a)(b)(c)(d)(e)1 4227 21627132x38x343. h(x) 13xDomain: (0, )r44. k(x) 13xDomain: (0, )(d) 16x2 12x 2(g) x2 11x 30(e) 4x2 12x 8(h) x2 3x 2(f) x2 3x 2(i) x4 3x2 22x322 3(g)(x 4)3x 12x2 48x 6422 4x3(h) 3 4 xx32(i) 6x(f) (f) 3a2 3a 1416412a2 6a 26a2 6a 43a2 15a 168. (a) f ( 4) 1(b) f ( 3) 29. (a) f (4) 4(g)(h)(i)12 a6 2a23a23a2 2 13a2 6ah 3h2(c) f (3) 0 3a 3h 2(e) f ( 3.001) 1.999(d) f (3.001) 1.999(f) f (2) 5(d) f (0) 1(b) f ( 3) 9(e) f ( 1) 1(c) f (1) 0(f) f ( 0.999) 0.0447101778Page 123
54Relations and Functions10. (a) ( , )(i) ( , )(b) ( , )(j) ( , 8) (8, )(c) ( , 6) ( 6, 6) (6, ) (d) 13 , (e) 13 , (k) [0, 8) (8, )(f) ( , ) (g) 31 , 3 (3, ) (h) 13 , 6 (6, )(l) (8, )(m) ( , 8) (8, )(n) [7, 9] (o) , 12 12 , 0 0, 12 12, (p) [0, 25) (25, )11. The applied domain of P is [0, ). The range is some subset of the natural numbers becausewe cannot have fractional Sasquatch. This was a bit of a trick question and we’ll address thenotion of mathematical modeling more thoroughly in later chapters. P (0) 0 means thatthere were no Sasquatch in Portage County in 1803. P (205) 139.77 would mean there were139 or 140 Sasquatch in Portage County in 2008.12. (a) b0.785c 0, b117c 117, b 2.001c 3, and bπ 6c 9Page 124
1.6 Function ArithmeticTaken from Stitz and Zeager4. Find and simplify the difference quotient(a)(b)(c)(d)(e)(f)f (x) 2x 5f (x) 3x 5f (x) 6f (x) 3x2 xf (x) x2 2x 1f (x) x3 12(g) f (x) x3f (x h) f (x)for the following functions.h3(h) f (x) 1 xx(i) f (x) x 9 3(j) f (x) x(k) f (x) mx b where m 6 0(l) f (x) ax2 bx c where a 6 0Rationalize the numerator. It won’t look ‘simplified’ per se, but work through until you can cancel the ‘h’.Page 12561
1.6 Function Arithmetic4. (a)(b)(c)(d)(e)(f)2 306x 3h 1 2x h 23x2 3xh h22(g) x(x h)ANSWERS633(1 x h)(1 x) 9(i)(x 9)(x h 9)1(j) x h x(k) m(h)(l) 2ax ah bPage 126
Domain of a FunctionFind the domain of the following (write answers in intervalnotation):5x 71.f (x ) 2xx 5x 614.f (x ) 2.f (x ) x9 x215.f (x ) x 2 43.3x 7f (x ) 2x 6x 2716.f ( x ) 12x 2 11x 517.f ( x ) x 2 5x 618.f ( x ) x 2 3x 419.f ( x ) x 2 2x 82x2 94.f (x ) x 88 x 2x 2 3 x5.f (x ) 4xx 256.f ( x ) 3x 520.f (x ) 9 x27.f (x ) x 521.f ( x ) 100 x 28.f ( x ) 3x 722.f ( x) 9.f ( x ) 12x 2423.f ( x ) 6x 2 x 1210.f ( x ) 9x 2724.f ( x ) 15x 2 4 x 311.f (x ) 5 x 212.1f (x ) 3x 125.f (x ) 26.f ( x ) 4 3x 1513.f (x ) 3212xx 5Page 127x2 4x 5x 1212
Domain of a Function-Answers1.( , 3 ) ( 3, 2 ) ( 2, )15.( , 2] [2, )2.( , 3 ) ( 3,3) (3, )16.3.( , 3 ) ( 3,9 ) ( 9, )5 1 , 4 3 , 17.( , 3] [ 2, )4.1 1 3 3 , 2 2 ,0 0, 4 4 , 18.( , 1] [4, )19.( , 2] [4, )5.( , 5 ) ( 5,5) (5, )20.[ 3,3]6.( , )21.[ 10,10]7.[ 5, )22.( , )8. 7 3 , 23.9.[2, )4 3 , 3 2 , 1 3 , 3 5 , 10.( ,3]24.11.( , )25.[5,11) (11, )26.[5, ) 1 12. , 3 13.(5, )14.( , 3 ) (3, )Page 128
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In exercises 21-38 determine the domain and range of each functions whose graph is given.Express your answers using interval notation.21.22.65432154321-4 -3 -2 -1-1-2-3-4-523.1 2 3 4-4 -3 -2 -1-1-2-324.543211 2 3 425.1 2 3 427.4321-4 -3 -2 -1-1-2-3-41 2 3 428.-4 -3 -2 -1-1-2-3-41 2 3 4-4 -3 -2 -1-1-2-3-41 2 3 429.432143211 2 3 426.432154321-4 -3 -2 -1-1-2-3-4-5-4 -3 -2 -1-1-2-3-4-5-4 -3 -2 -1-1-2-3-443211 2 3 4Page 141-4 -3 -2 -1-1-2-3-41 2 3 4
30.31.43214321-4 -3 -2 -1-1-2-3-432.1 2 3 4-4 -3 -2 -1-1-2-3-432-4 -3 -2 -1-1-234.335.11 2 3 4-4 -3 -2 -1-1-2-2-336.543215432121 2 3 437.54321-4 -3 -2 -1-1-21 2 3 4-333.-4 -3 -2 -1-111 2 3 41 2 3 4-4 -3 -2 -1-1-21 2 3 438.332211-4 -3 -2 -1-11 2 3 4-2Page 142-4 -3 -2 -1-1-21 2 3 4
In exercises 39-40, use the graphs to determine the intervals where each function is increasing,decreasing, or constant. Express your answers using interval 2-4-4-6Sketch the following piecewise functions: 2x 1 if x 0 2x 1 if x 241. f ( x ) 49. f ( x ) if x 2 5 x if x 0 5 3 x42. f ( x ) x 2if x 0if x 0 x50. f ( x ) 32 x 1 if x 2 x 243. f ( x ) 51. f ( x ) 3x 9 if x 2 4if x 0if x 0if x 1if x 1 2x 5 if x 0 if 0 x 454. f ( x ) 5 x 2if x 4 1 if x 1 55. f ( x ) xif 1 x 1 1if x 1 x 1 if x 2 156. f ( x ) 3if 2 x 3 2 x 5 if x 2 2x 9 if x 352. f ( x ) 3x 7 if 2 x 3 3x if x 1 x 145. f ( x ) 2if x 3 2x 4 if x 1 xifx 1 57. f ( x ) 2if 1 x 2 x 4 if x 2 4 if x 2 x 6 if x 446. f ( x ) 2 xifx 2 153. f ( x ) x 7 if 4 x 2 2 3x 5 if x 247. f ( x ) if x 2 x 5 1 if x 2 3 x if x 244. f ( x ) 3x 5 if x 2 x 4 if x 148. f ( x ) if x 1 1Page 143
Section 3.5-Interpreting Graphs1. 327.3. 35. 129.7. 29. 431.11. Yes, it is a functionD ( , )R [ 4, )D [ 3, 3]R [ 1, 2]D ( , )R {All integers}13. No, not a function15. Yes, it is a function33.17. Yes, it is a function19. No, not a function21.23.25.D ( , )35.R [ 4, )D ( , )37.R [ 0, )39.R ( , ) 2x 1 if x 0 5 x if x 0R 1 [ 0, )D ( , )R ( , 2) ( 2, )D [ 0, 4]R [ 0, 2]( 4, 1)Intervals of Decreasing: ( 2, )Intervals of Constant: ( , 4 ) ( 1, 2)Intervals of Increasing:D ( , )41. f ( x ) D ( , ) x 1 if x 2 3x 9 if x 243. f ( x ) Page 144 3x45. f ( x ) x2if x 1if x 1
3x 5 if x 2 1 if x 247. f ( x ) x 6 if x 4 153. f ( x ) x 7 if 4 x 2 2if x 2 x 5 2x 1 if x 2if x 2 549. f ( x ) 1 if x 1 55. f ( x ) xif 1 x 1 1if x 1 2x 4 if x 1 if 1 x 257. f ( x ) 2 x 4 if x 2 Page 145 x 251. f ( x ) 4if x 1if x 1
1.7 Graphs of Functions73Example 1.7.4. Given the graph of y f (x) below, answer all of the following questions.y4(0, 3)321( 2, 0) 4 3 2(2, 0) 11234x 1 2 3( 4, 3)(4, 3) 49. List the intervals on which f is increasing.1. Find the domain of f .2. Find the range of f .10. List the intervals on which f is decreasing.3. Determine f (2).11. List the local maximums, if any exist.4. List the x-intercepts, if any exist.5. List the y-intercepts, if any exist.12. List the local minimums, if any exist.6. Find the zeros of f .13. Find the maximum, if it exists.7. Solve f (x) 0.14. Find the minimum, if it exists.8. Determine the number of solutions to theequation f (x) 1.15. Does f appear to be even, odd, or neither?AdditionalSolution. problems taken from Stitz and Zeager1. To find assignment:the domain of f , p.we 73:proceedas in Section 1.4.Suggested1-10By projecting the graph to the x-axis,we see the portion of the x-axis which corresponds to a point on the graph is everything from 4 to 4, inclusive. Hence, the domain is [ 4, 4].2. To find the range, we project the graph to the y-axis. We see that the y values from 3 to3, inclusive, constitute the range of f . Hence, our answer is [ 3, 3].3. Since the graph of f is the graph of the equation y f (x), f (2) is the y-coordinate of thepoint which corresponds to x 2. Since the point (2, 0) is on the graph, we have f (2) 0.Page 146
1.7 Graphs of Functions771.7.2 ExercisesAdditionalproblems taken from Stitz and Zeager1. Sketch the graphs of the following functions. State the domain of the function, identify anyinterceptsand test for symmetry.Suggestedassignment:p. 77: 3 (a-j)(a) f (x) x 23(b) f (x) 5 x(c) f (x) 3x(d) f (x) 1x2 12. Analytically determine if the following functions are even, odd or neither.(a) f (x) 7x(b) f (x) 7x 2(c) f (x) 1x3(d)(e)(f)(g)(h) f (x) x4 x3 x2 x 1 (i) f (x) 5 xf (x) 4f (x) 0f (x) x6 x4 x2 9f (x) x5 x3 x(j) f (x) x2 x 63. Given the graph of y f (x) below, answer all of the following questions.y54321 5 4 3 2 1 112345x 2 3 4 5(a)(b)(c)(d)(e)(f)(g)(h)Find the domain of f .Find the range of f .Determine f ( 2).List the x-intercepts, if any exist.List the y-intercepts, if any exist.Find the zeros of f .Solve f (x) 0.Determine the number of solutions to theequation f (x) 2.(i) List the intervals where f is increasing.(j) List the intervals where f is decreasing.(k) List the local maximums, if any exist.(l) List the local minimums, if any exist.(m) Find the maximum, if it exists.(n) Find the minimum, if it exists.(o) Is f even, odd, or neither?Page 147
1.7 Graphs of Functions—Stitz and Zeager BookANSWERS p. 73:1-15[ 4, 4][ 3, 3]f ( 2) 0( 2, 0),( 2, 0)( 0, 3)x 2, 2[ 4, 2] ( 2, 4]8. 2 solutions9. [ 4, 0)10. ( 0, 4]11. ( 0, 3)12. none13. 314. 315. yes, even1.2.3.4.5.6.7.p. 77: 3 (a-j) ANSWERSPage 148
3.6—Additional Graphing TechniquesIn problems 1-40 use the techniques of shifting, reflecting, and stretching to sketch thegraph of the following functions.1. f ( x) ( x 1) 2 32. f ( x) ( x 1) 2 43. f ( x) x 1 24. f ( x) x 2 15. f ( x) x 3 26. f ( x) 2 x 2 17. f ( x) x 4 48. f ( x) 2 x 3 39. f ( x) 3 x 2 310. f ( x) 3 x 1 211. f ( x) 2 x 1 112. f ( x) 2 x 3 213. f ( x) 3 x 214. f ( x) 3321 2x416. f ( x) x 115. f ( x) x 317. f ( x) x 4119. f ( x) x 3 3221. f ( x) x 1 323. f ( x) 2 x 1 2313x 2 12227. f ( x) 2 x 3 525. f ( x) 29. f ( x) 2 3 x 1 2131. f ( x) x 3 221333. f ( x) x 1 3235. f ( x) 3 x 2 118. f ( x) x 1120. f ( x) x 2 12122. f ( x) x 1 4224. f ( x) x 326. f ( x) x 3 313x 1 4230. f ( x) 2 3 x 328. f ( x) 32. f ( x) 2 x 1 134. f ( x) 1x 4 2237. f ( x) x 4 113 x 4 1238. f ( x) 3 x 3 139. f ( x) 2 x 3 140. f ( x) x 5 3336. f ( x) 3Page 162
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Section 3.6-Graphing Techniques1.3.5432154321-4 -3 -2 -1-1-2-3-4-57.1 2 3 413.-4 -3 -2 -1-1-2-3-4-59.7654321-2 -1-1-2-31 2 3 4 5 61 2 3 41 2 3 41 2 3 4-1-1-2-1-1-2543211 2 3 454321-4 -3 -2 -1-1-2-3-4-51 2 3 423.876543211 2 3 4 5 6 71 2 3 4 5 6-4 -3 -2 -1-1-2-3-4-517.432121.87654321-2 -1-1-2-3-4-511.54321-4 -3 -2 -1-1-2-3-41 2 3 419.54321-4 -3 -2 -1-1-2-3-4-515.54321-4 -3 -2 -1-1-2-3-4-55.43211 2 3 4 5 6 7Page 167-6 -5 -4 -3 -2 -1-1-2-3-41 2
25.27.4321-4 -3 -2 -1-1-2-3-4-51 2 331.29.1-2 -1-1-2-3-4-5-6-7-81 2 3 4 5 6-4 -3 -2 -1-1-2-3-4-533.1 2-4 -3 -2 -1-1-237.35.76543214321-6 -5 -4 -3 -2 -1-1-2-3-41 2 339.-2-3-441ii.-6-5-4-3-2-1-1 1 2 3 4 5 6-2-3-4-5-64321-2 -1-1-2-3-4-543211 2 3 4 5 6-3 -2 -1-16543211 2 3 4 5 6-2-3-441iii.6543211 2 341i.4321-8 -7 -6 -5 -4 -3 -2 -1-1 1 24321-6-5-4-3-2-1-1 1 2 3 4 5 6-2-3-4-5-641iv.654321-6-5-4-3-2-1-1 1 2 3 4 5 6-2-3-4-5-6Page 168654321-6-5-4-3-2-1-1 1 2 3 4 5 6-2-3-4-5-6
41v.41vi.41vii.654321654321-6-5-4-3-2-1-1 1 2 3 4 5 6-2-3-4-5-641viii.-6-5-4-3-2-1-1 1 2 3 4 5 6-2-3-4-5-643i.4321-1 1 2 3-9-8-7-6-5-4-3-2-1-2-3-4-5-6-7-8-943iii.-4 -3 -2 -1-1 1 2 3 4 5 6 7 8-2-3-4-6-5-4-3-2-1-1 1 2 3 4 5 6-2-3-4-543ii.654321654321-4 -3 -2 -1-1 1 2 3 4 5 6 7 8-2-3-443iv.65432154321-4 -3 -2 -1-1 1 2 3 4 5 6 7 8-2-3-443v.4321-4 -3 -2 -1-1 1 2 3 4 5 6 7 8-2-3-4-5-6Page 169654321-4 -3 -2 -1-1 1 2 3 4 5 6 7 8-2-3-4
43vi.43vii.6543214321-6 -5 -4 -3 -2 -1-1 1 2 3 4 5 6-2-3-445. even47. neither43viii.-6 -5 -4 -3 -2 -1-1 1 2 3 4 5 6-2-3-4-5-649. odd51. neither67.53. odd55. even63. even65. odd71.254321543211 2 3 473.1-4 -3 -2 -1-1-21 2 3 477.5432143211 2 3 4-1-275.-4 -3 -2 -1-1-2-3-4-4 -3 -2 -1-1 1 2 3 4 5 6 7 8-2-3-457. even61. neither69.-4 -3 -2 -1-1-2654321-4 -3 -2 -1-1-2-3-4-543211 2 3 4Page 170-4 -3 -2 -1-1-2-3-41 2 3 4
104Taken from Stitz and Zeager1.8.1ExercisesRelations and Functions1. The complete graph of y f (x) is given below. Use it to graph the following functions.y(0, 4)4321 4 3 1( 2, 0) 1 2134x(2, 0)(4, 2) 3 4The graph of y f (x)(a) y f (x) 1(d) y f (2x)(g) y f (x 1) 1(b) y f (x 1)(e) y f (x)(h) y 1 f (x)(f) y f ( x)(i) y 12 f (x 1) 1(c) y 12 f (x)2. The complete graph of y S(x) is given below. Use it to graph the following functions.y(1, 3)321( 2, 0) 2(0, 0) 11(2, 0)x 1 2 3( 1, 3)The graph of y S(x)(a) y S(x 1)(c) y 21 S( x 1)(b) y S( x 1)(d) y 12 S( x 1) 1Page 171
1.8 Transformations1053. The complete graph of y f (x) is given below. Use it to graph the following functions.y3(0, 3)21 3 2 1( 3, 0)1(c) j(x) f x 3x(g) d(x) 2f (x) (h) k(x) f 32 x(a) g(x) f (x) 3(b) h(x) f (x) 2(3, 0) 112 23(i) m(x) 14 f (3x)(d) a(x) f (x 4)(j) n(x) 4f (x 3) 6(e) b(x) f (x 1) 1(k) p(x) 4 f (1 2x) (l) q(x) 12 f x 4 32(f) c(x) 35 f (x) 4. The graph of y f (x) 3 x is given below on the left and the graph of y g(x) is givenon the right. Find a formula for g based on transformations of the graph of f . Check youranswer by confirming that the points shown on the graph of g satisfy the equation y g(x).yy554433221 11 10 9 8 7 6 5 4 3 2 1 1112345678x 11 10 9 8 7 6 5 4 3 2 1 1 2 2 3 3 4 4 5 5 y 3x12345678xy g(x)5. For many common functions, the properties of algebra make a horizontal scaling the sameasscaling by (possibly) a different factor. For example, we stated earlier that a vertical 9x 3 x. With the help of your classmates, find the equivalent vertical scaling produced 2by the horizontal scalings y (2x)3 , y 5x , y 3 27x and y 12 x . What about 2 y ( 2x)3 , y 5x , y 3 27x and y 12 x ?Page 172
1.8 Transformations1.8.2107Answers1. (a) y f (x) 1(d) y f (2x)yy4(0, 4)43(0, 3)3221(1, 0)1 4 3 2 11 2 1( 2, 1)(2, 1) 234 4 3 2 ( 1, 0)x2 234x(2, 2) 3 3(4, 3) 4 4(e) y f (x)y(b) y f (x 1)y( 1, 4)443322( 2, 0)(4, 2)1(2, 0)1 4 3 2 1 1( 3, 0) 21 2(1, 0)34 4 3 2 1 1x12343(2, 0)4x 2 3(3, 2) 3(0, 4) 4 4(f) y f ( x)(c) y 12 f (x)yy44332(0, 4)2(0, 2)11 4 3 4 3 1( 2, 0) 1 2134x(2, 0) (4, 1)( 4, 2) 1( 2, 0) 1 2 3 3 4 4Page 1731x
108Relations and Functions(i) y 12 f (x 1) 1(g) y f (x 1) 1y( 1, 3)y4433221( 1, 1) 4 3 2 1 112 3 4(1, 1)x 4 3 2 1 1( 3, 1) 2 2( 3, 1) 3112 3 4(1, 1)(3, 2) 3(3, 3) 4 4(h) y 1 f (x)y4(4, 3)32( 2, 1)(2, 1)1 4 3 2 1 11234x 2(0, 3) 3 4(b) y S( x 1)2. (a) y S(x 1)y3( 3, 0) 3y(0, 3)32211(0, 3)( 1, 0) 2(1, 0) 1(1, 0)x 1( 1, 0) 1 2 2 3 3( 2, 3)12(2, 3)Page 174(3, 0)3xx
1.8 Transformations109(c) y 21 S( x 1)(d) y 12 S( x 1) 1yy230,32 0,152 2(1, 0)1( 1, 0) 1(3, 0)2(1, 1)x3 12, 23 2 1x32, 12 1 (d) a(x) f (x 4)3. (a) g(x) f (x) 3y6(3, 1)1( 1, 1)y( 4, 3)(0, 6)352413( 3, 3) 7 6 5 4 3 2 1( 7, 0)( 1, 0)(3, 3)21(e) b(x) f (x 1) 1 3 2 1123yx( 1, 2) 12(b) h(x) f (x) y3x112 4 3 2 1 50, 212x 1( 4, 1)(2, 1)2(f) c(x) 35 f (x)1 3 2 1 3, 1 121y 2 3 x3, 1220, 95 1 3 2 1( 3, 0) 1(c) j(x) f x 23y 2,33321 3 2 1 7,0 13123 x11 , 03Page 175123(3, 0)x
110Relations and Functions(g) d(x) 2f (x)(j) n(x) 4f (x 3) 6yy( 3, 0)(3, 0)(3, 6)6 3 2 112x3 15 24 33 42 51 61(0, 6)23456x 1 2 3 4 5 6(h) k(x) f23x(6, 6)(0, 6) y(k) p(x) 4 f (1 2x) f ( 2x 1) 4(0, 3)3y 271 1,726 4 3 2 1 9,0 12123 45x 9,024( 1, 4)3(i) m(x) 14 f (3x)y(2, 4)21( 1, 0)(1, 0) 1 1x112x 1 10, 34 (l) q(x) 21 fx 42 3 12 fy 10 9 8 7 6 5 4 3 2 1 1 2( 10, 3) 3 4 9 4, 2 4. g(x) 2 3 x 3 1 or g(x) 2 3 x 3 1Page 17612x12x(2, 3) 2 3
Piecewise FunctionsGraph the following: 2x1. f ( x ) 1if x
Absolute Value Inequalities Solve the inequalities. Graph your answer on a number line. Write answers in interval notation. 1. x 3 2. x 5 3. x 27 4. 3510x 5.
