MTH101_Chap2_4 - A: Functions 59 (D) 5(E) 200 g I [5 g Y .5...

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Unformatted text preview: A: Functions 59 (D) 5(E) 200 g I [5 g Y .5 E —200 Million cameras E xerc15e 2-1 A In Problems 1—8, use point-by-point plotting to sketch the 17- graph of each equation. 1. y = x + 1 2. x = y + 1 3. x = y2 4. y = x2 5. y = x3 6. x = y1 7- xY = ‘6 19. .v 20. .v ‘ 8' XY = [Wj'ttj’ +T1l1 *3: h Indicate whether each table in Problems 9—14 specifies a function. 77:55:” f 71“ ‘ l »-——-—--——’——-—-—'— W . 'V r4 I . L‘L: , ‘-:A' ,4 fibre—j 9. Domain Range 10. Domain Range l ‘ I ‘_ L» It so “ ..._._._._._____.___.. -—-———-—-—-—-—-————-—‘ ' x , A A ‘ A‘ x _,, . _._._...___. ._ ________. s , ‘ ~ other , a z ‘3 _; 7 firm» i?" _____. ....___+ , 4'? 7 _ 7—————~——+ 2 ~3—————————. 9 lie-+1 : :41 :L 1‘} l: _ *i l 11. Domain Range 12. Domain Range In Problems 21-30, each equation specifies a function. " Determine whether the function is linear, constant, or i 3 < z 5 8 ' 0 neith r. t 6 9 i l e I 4 .4: 7 2 21. y = 3 — 7x 5 -——————» 8 10 —————. 3 22. y = 1,. 23. y = ——6 “AW ~—-——-——-—-—-——- _ _ l _ 13. Domain Range 14. Domain Range 24‘ y _ 5x 2(4 x) l _-_ _ .. .. ____..____ ~———-—————-——————— 1 . = — + — 3 \ 5 ~2 \ 25 y 8x 1 x i 6 1/” —t \ __ 1 K /5 26. y—3x+§(5~—6x) 9 \ 6 0 / 1 + x 1 — x 1 '2 f l 27. y = 2 + 3 a - Indicate whether each graph in Problems 15—20 specifies a 28' y = 1 flmction. 2x + 3 1 15. y 16. .v 29. y = x2 + (1 — x)(1 + x) + 1 l ., - ' 30.y=x2—9 In Problems 31 and 32 which of the indicated correspondences X define functions? Explain. 31. Let P be the set of residents of Pennsylvania and let R and S be the set of members of the US. House of Representatives and the set of members of the US. Senate, respectively, elected by the residents of Pennsylvania. 60 C H A PT E R 2 Functions and Graphs (A) A resident corresponds to the congressperson rep— 45. y = f (—5) 46. y = f(4) resenting the resident’s congressional district. ; 47. y ___ “5) 48. y = f(_2) (B) A resident corresponds to the senator representing 49 O = f(x) 50 3 = f(x) x < 0 the resident’s state. ' ‘ ' 51. -4 = f(x) 52. 4 = f(x) 32. Let P be the set of patients in a hospital, let D be the set of doctors on the hospital staff, and N be the set of nurses on the hospital staff. If f(x) = 2x — 3 and g(x) = x2 + 2x, find each of the expres- . , sions in Problems 53—64. (A) A patient corresponds to the doctor if that doctor f . C admitted the patient to the hospital. 5 53' f0) 54' K1) , (B) A patient corresponds to the nurse if that nurse 55- f(‘ll 56- 8(1) \ cares for the patient. 57. g(_3) 58. g(_2) In onbllcemsi3fil—40;.use point-by-point plotting to sketch the 59. 100) + ga) 60_ f(3) _ g(3) 1 ra o eac nc ton. ‘ r g p x g 61. 3(3)'f(0) 62. g(O)-f(-2) r 33. f(x)=1-x 34. f(x)=--3 l __ ‘ 2 32 2 63 8( Z) 64' g(—3) ‘ 35. f (x) — x — 1 36. f(x) — - x f(_2) _f'—(2)" ‘ 37. f(x) = 4 — x3 38. f(x) = x3 — 2 39. f(x) = § 40' f(x) = "_6 B In Problems 65—70, find the domain of each fimction. E x x 65. F(x)=2x3—x2+3 66.1-l(x)=7—2x2—x4 3 In Problems 41 and 42, the three points in the table are on the x _ 2 x + 1 l graph of the indicated function f Do these three points provide 67' f(x) : x + 4 68' g“) = x _ 2 ' sufiicient information for you to sketch the graph of y = f (x)? 1 l ; . = V7 - 7 . = --———- . Add more points to the table until you are satisfied that your 69 got) x 0 F“) 5 + x ‘ sketch is a good representation of the graph of y = f (x) on the 71. Two people are discussing the function ‘ Interval [-5, 5]. g i i x2 — 4 f = 2 f(x) — 2" l x — 9 r _ 2 x + 1 and one says to the other, “f(2) exists but f(S) does not.” 1 3 Explain what they are talking about. . 3x2 72. Referring to the function in Problem 71, do f (—2) and 1 f(x) = x2 + 2 f (—3) exist? Explain. 43_ Let f(x) = 100x _ 5x2 and g(x) = 150 + 20x, The verbal statement “function f multiplies the square of the 0 S x S 20 domain element by 3 and then subtracts 7 from the result" and x ' .y the algebraic statement "f(x) = 3x2 — 7” define the same (A) Evaluate f(x),g(x), and f(x) — g(x) for x = 0, 5, function. In Problems 73—76, translate each verbal definition l 10, 15, 20. j of a function into an algebraic definition. } (3) Graph y = f(x), y = got), and y = f(x) _ 30‘) on 1 73. Function g subtracts 5 from twice the cube of the domain the interval [0,20]. ; element } 44 Repeat Problem 43 for f(x) = 200x __ 10x2 and 5 74. Function f multiplies the domain element by -3 and i g(x) = 200 + 50L adds 4 to the result. 3 ’ 75. Function G multiplies the square root of the domain ele— ‘ In Problems 45—52, use the following graph of a function f to ; ment by 2 and subtracts the square of the domain ele- determine x or y to the nearest integer, as indicated. Some prob- , ment from the result. \ [ems may have more than one answer‘ i 76. Function F multiplies the cube of the domain element ‘ f(x) y = f(x) by —8 and adds 3 times the square root of 3 to the result. In Problems 77—80, translate each algebraic definition of the function into a verbal definition. 77. f(x) = 2x — 3 78. g(x) = —2x + 7 79. F(x) = 3x3 — 2v; so. G(x) = 4v; — x2 Determine which of the equations in Problems 81—90 specify ; functions with independent variable x. For those that do, find l l the domain. For those that do not, find a value of .r to which there corresponds more than one value of y. 81. 4x—5y=20 82. 3y-7x=15 83. x2-y=' 84.x—y2= 85.x+y2=10 86.x2+y=10 87.xy—4y= 88.xy+y-x= 89.x2+y2=25 90.x2-y2=l6 91. If F(t) = 4t + 7, find F(3 + h) — F(3) h 92. If C(r) = 3 — 5r,£ind Section 2~t Functions 61 C In Problems 107.112, find and simplify each of the following. (A) f(x + h) (B) fix + h) r f(x) (C) f(x + h]: r fix) 107. f(x) = 4x ~ 3 108. f(x) = -3x + 9 109. f(x) = 4x2 - 7x + 6 110. f(x) = 3x2 + 5x — 8 111. f(x) = x(20 — x) 112. f(x) = x(x + 40) Problems 113—116, refer to the area A and perimeter P of a rec- tangle with length [and width to (see the figure). 93. If f(x) = x2 ~ 1, find and simplify each expression in Prob- G(2 + h) — 0(2) h IfQ(x) = x2 — 5x + 1, find Q(2 + h) — (2(2) h , if P(x) = 2x2 — 3x — 7, find I i w ' 113. The area of a rectangle is 25 square inches. Express the perimeter P(w) as a function of the width w, and state the domain of this function. [ems 95405 114. The area of a rectangle is 81 square inches Express the 95. 97. 99. 101. 103. 105. perimeter F(l) as a function of the length l, and state the f(5) 95- “*3 domain of this function. f (2 + 5) 98- f (3 ' 5) 115. The perimeter of a rectangle is 100 meters. Express the fa) + “5) 100. “3) ._ “6) area A(l) as a function of the length l, and state the do~ main of this function. f(f(1)) 102- f(f(—2)) . . fax) 1% f(_3x) 116. The perimeter of a rectangle IS 160 meters. Express the area A(w) as a function of the Width 10, and state the do- f(x + 1) 106. f(1 -— x) main of this function. Applications \117..3Price-—demand. A company manufactures memory chips for g 118. Price—demand. A company manufactures “notebook” com- microcomputers Its marketing research department, using sta- ‘ tistical techniques, collected the data shown in Table 8, where 3 p is the wholesale price per chip at which x million chips can ‘ be sold. Using special analytical techniques (regression analy- sis), an analyst produced the following price—demand func- tion to model the data: Plot the data points in Table 8, and sketch a graph of the price- demand function in the same coordinate system. What would ‘ be the estimated price per chip for a demand of 7 million . chips? For a demand of 11 million chips? puters. Its marketing research department, using statistical techniques, collected the data shown in Table 9, where p is the wholesale price per computer at which A: thousand computers can be sold. Using special analytical techniques (regression analysis). an analyst produced the following price—demand \ ‘ function to model the data: p(x)=75—3x lsxSZO p(x)=2,000~60x lsxs25 TABLE 9 Prite»Demand TABLE 8 Price Demand x (Millions) p (s) 1 (Thousands) p (S) 1 72 1 l ,940 4 63 3 1,520 9 48 15 1,040 14 33 21 740 20 15 25 500 Plot the data points in Table 9, and sketch a graph of the price—demand function in the same coordinate system. What would be the estimated price per computer for a demand of ll thousand computers? For a demand of 18 thousand computers? 62 C H A P T E R 2 Functions and Graphs @ Revenue. (A) Using the price—demand function p(x)=75—3x 1.<_xs20 (C) Plot the points in part (B) and sketch a graph of the profit function through these points. 122. Profit. The financial department for the company in Prob— from Problem 117, write the company’s revenue function and indicate its domain. (B) Complete Table 10, computing revenues to the nearest million dollars TABLE 10 Rumour» I (Millions) R(x) (Million 3) 1 , 72 4 s 12 16 20 (C) Plot the points from part (B) and sketch a graph of the revenue function through these points Choose millions for the units on the horizontal and vertical axes 120. Revenue. (A) Using the price—demand function p(x)=2,000—60x leSZS from Problem 118, write the company’s revenue function and indicate its domain. (B) Complete Table 11, computing revenues to the nearest thousand dollars. TABLE 11 wat‘mit} x (Thousands) R(x) (Thousand S) 1. ' 1.940 ,5 p , 10 15 20 i 25: (C) Plot the points from part (B) and sketch a graph of the revenue function through these points. Choose thou- sands for the units on the horizontal and vertical axes. 121. rofit. The financial department for the company in oblems 117 and 119 established the following cost func- tion for producing and selling x million memory chips: C(x) = 125 + 16x million dollars (A) Write a profit function for producing and selling x million memory chips, and indicate its domain. (B) Complete Table 12, computing profits to the nearest million dollars. TABLElZ awn» x (Millions) P(x) (Million 3) ‘ 1 i ”‘ ~69 . .8 V (12', I16 f, 20 lems 118 and 120 established the following cost function for producing and selling x thousand “notebook” computers: C(x) = 4,000 + 500;: thousand dollars (A) Write a profit function for producing and selling x thousand f‘notebook” computers, and indicate the domain of this function. (B) Complete Table 13, computing profits to the nearest thousand dollars YABLE 13 Built 1: (Thousands) P(x) (Thousand S) 1 - - ram 5 _, _ 10 15 20 25 (C) Plot the points in part (B) and sketch a graph of the profit function through these points. 123. Packaging. A candy box is to be made out of a piece of carc board that measures 8 by 12 inches. Equal-sized squares x inches on a side will be cut out of each corner, and then the ends and sides will be folded up to form a rectangular box. (A) Express the volume of the box V(x) in terms of x. (B) What is the domain of the function V (determined by the physical restrictions)? (C) Complete Table 14. TABlEM ‘ 2 MW“ x W1) , ., , - r .g' ' r ,fi . 2 3 (D) Plot the points in part (C) and sketch a graph of the volume function through these points. 124. Packaging. Refer to Problem 123. (A) Table 15 shows the volume of the box for some value: of 1: between 1 and 2. Use these values to estimate to or TABLE 15 x v1» 14 “r ‘7,, ~ 623%: L2 ' ' ; r,' 64512 13" r 3- ,- Gina 14' T _ . soon 15; r _ » ~61; ,15 ,t, ," J .s.'61§M 13- ‘ 612% ~ decimal place the value of x between 1 and 2 that would produce a box with a volume of 65 cubic inches. i; 2%) Describe how you could refine this table to estimate x to two decimal places (C) Carry out the refinement you described in part (B) and approximate x to two decimal places. 125. Packaging. Refer to Problems 123 and 124. (A) Examine the graph of V(x) from Problem 123D and discuss the possible locations of other values of x that would produce a box with a volume of 65 cubic inches. Construct a table like Table 15 to estimate any such value to one decimal place. (B) Refine the table you constructed in part (A) to provide "i an approximation to two decimal places. 126. Packaging. A parcel delivery service will only deliver pack- ages with length plus girth (distance around) not exceeding 108 inches. A rectangular shipping box with square ends x inches on a side is to be used. Length 4 (A) If the full 108 inches is to be used, express the volume of the box V(x) in terms of x. (B) What is the domain of the function V (determined by the physical restrictions)? 127. 128. Elementary Functions: Graphs and Transformations 63 (C) Complete Table 16. .r le) ll) 20 25 (D) Plot the points in part (C) and sketch a graph of the volume function through these points Muscle contraction. In a study of the speed of muscle con- traction in frogs under various loads, noted British biophysi- cist and Nobel Prize winner A. W. Hill determined that the weight w (in grams) placed on the muscle and the speed of contraction v (in centimeters per second) are approximately related by an equation of the form (w+n)('v+b)=c where a, b, and c are constants Suppose that for a certain mus- cle, = 15, b = 1, and c = 90. Express 1) as a function of w. Find the speed of contraction if a weight of 16 grams is placed on the muscle. Politics The percentage 3 of seats in the House of Represen- tatives won by Democrats and the percentage 1) of votes cast for Democrats (when expressed as decimal fractions) are re- lated by the equation 50—25=1.4 0<s<1, O.28<’u<0.68 (A) Express 1) as a function of s, and find the percentage of votes required for the Democrats to win 51% of the seats (B) Express 5 as a function of v, and find the percentage of seats won it Democrats receive 51 % of the votes Section 2-2 ELEMENTARY FUNCTIONS: GRAPHS AND TRANSFORMATIONS a A Beginning Library of Elementary Functions I Vertical and Horizontal Shifts I Reflections, Stretches, and Shrinks n Piecewise—Defined Functions The functions go) : x2 — 4 h(x) = (x — 4)2 k(x) = —4x2 all can be expressed in terms of the function f(x) = x2 as follows: glx) = flx) ~ 4 h(x) = f(x * 4) k(x) = ‘4f(x) In this section we will see that the graphs of functions g, h, and k are closely related to the graph of function f. Insight gained by understanding these relationships will help us analyze and interpret the graphs of many different functions Elementary Functions: Graphs and Transformations 73 A wit/tout looking back in the text, indicate the domain and range 31. 32. ofcaclt oft/1e functions in Problems [—8. 1. f(x) = 2x 2. g(x) = —O.3x 3. h(x) = —0.6\/} 4. k(x) = 4v; 5. m(x) = Slxl 6. n(x) = —0.Ix2 7. r(x) = —x3 8. s(x) = 5% Graph each of the functions in Problems 9—20 using the graphs of functions f and g below. gm 9. y=f(x)+2 10. y=g(.r)—l 11. y = f(x + 2) 12. y = g(x — I) 13. y = g(.r — 3) 14. y = f(x + 3) i 15. y = g(.r) — 3 16. y = f(x) + 3 17. y = —f(x) 18. y = -g(.t) 19. y = 0.5g(x) 20. y = 2f(x) B In Problems 21—28, indicate verbally how the graph of each function is related to the graph of one of the six basic functions in Figure 1 on page 65. Sketch a graph of each function. 30:) = -|x + 31 h(x) = —lx — 5] 4f(x)=(x-4)2-3 ,, m(x) = (x + 3)2 + 4 2 f(x) = 7 — V} . g(x) = -6 + \V; _ I h _ In Problems 37—42, the graph of the function g is formed by ’ (x) _ ~3lx| applying the indicated sequence of transformations to the given " m(x) = ‘0-4x2 function )3 Find an equation for the function g and graph g ; Each graph in Problems 29—36 is the result of applyinga sequence "Sing "5 S x S 5 and '5 s y S 5- of transformations to the graph of one of the six basic functions in 37. The graph of f (x) = V} is shifted 2 units to the right Figure I on page 65. Identify the basic function and describe the and 3 units down, transformation verbally. Write an equation for the given graph 38. The graph of f(x) = V; is Shifted 3 units to the le ft and 2 units up. 39. The graph of f(x) = lxl is reflected in the x axis and shifted to the left 3 units. 40. The graph of f (x) = Ix] is reflected in the x axis and shifted to the right 1 unit. 41. The graph of f (x) = x3 is reflected in the x axis and shifted 2 units to the right and down 1 unit. 42. The graph of f (x) = x2 is reflected in the x axis and shifted to the left 2 units and up 4 units 74 C H A P T E R 2 Functions and Graphs Graph each function in Problems 43—48. 2—2): ifx<2 43-f(x)={x—2 ifoZ x-l'l ifx<‘1 44' 8U)" 2+2x ifo—l 5+0.5x imsxle 4S. h(x)—{_10+2x 10+2x ifOSxSZO 46. h(I)_{40+0.5x ifx>20 2x it05x520 47. h(x)= x+20 if20<x540 3 0.5x+45 ifx>40 i 4x+20 if0_<_x_<.20 48. h(x)= 2x+60 if20<x$100 -x+360 ifx>100 C Each of the graphs in Problems 49—54 involves a reflection in the 1 x axis and/or a vertical stretch 0r shrink of one of the basic fimc- é ' l "'0’? in Figure I 0" Page 65- Identify the basjcfi‘".di0": and d?‘ Changing the order in a sequence of transformations may scrlbzlhe ’ramfomatlo’l verbally WW9 0" “IMHO” for the 8W9" change the final result. Investigate each pair of transformations EMF 6 in Problems 55—60 to determine if reversing their order can produce a different result. Support your conclusions with spe- cific examples and/or mathematical arguments. 55. Vertical shift; horizontal shift 50. 56. Vertical shift; reflection in y axis 57. Vertical shift; reflection in x axis i \ 58. Vertical shift; vertical stretch l 1, 59. Horizontal shift; reflection in y axis 60. Horizontal shift; vertical shrink .1 . Applications 9 61. Price-demand. A retail chain sells CD players. The retail 63. Hospital costs. Using statistical methods, the financial 1; price p(x) (in dollars) and the weekly demand x for a par— department of a hospital arrived at the cost equation - r 1 d 1 1 db mu am" e are re are y ; C(x) = 0.000480: - 500)3 + 60,000 100 s x 5 1,000 p(x)=115—4\/J_l 92x5289 i (A) Describe how the graph of function F can be 0]} where C(x) is the cost in dollars for handling x cases per , , _ month. tamed from the graph of one of the basrc functions : in Figure 1 on page 65. (A) Describe how the graph of function C can be (B) Sketch agraph of function p using part (A) as an aid. Ob‘ained from the graph 0f one 0f the baSic functions in Figure 1 on page 65. % (B) Sketch a graph of function C using part (A) and a graphing calculator as aids. 62. Price—supply. The manufacturers of the CD players in Problem 61 are willing to supply x players at a price of p(x) as given by the equation P(x) = 4V3? 9 5- x 5 289 64. Price—demand. A company manufactures and sells in-line (A) Describe how the graph of function p can be ob. skates. Its financial department has established the tained from the graph of one of the basic functions price-demand function in Figure 1 on page 65. (B) Sketch a graph of function p using part (A) as an aid. 2 p(x) = 190 — 0.013(x — 10)2 10 s x s 100 Elementary Functions: Graphs and Transformations 75 where p(.r) is the price at which .r thousand pairs of skates can be sold. TABLE 6 Kansas State Income lax ' ‘ . SCHEDULE lI——SlN(iLE. HEAD OF HOUSEHOLD. 0R ) Describe how the graph of tunctlon p can be ob- M ARMED HUNG SEPARATE tained from the graph of one of the basic functions in Figure l on page'65. If taxable income is Over But Not Over Tux Duels (Bl SketCh a graph or funCtlon P “5mg Part (A) and a $0 $l5.000 3.50% of taxable income graphing calculator 35 NS- slslxil swim $525 plus 6.25% of 65. Electricity rates. Table 3 shows the electricity rates charged excess "vcr $15000 by Monroe Utilities in the summer months. The base is a 53mm ' “462-5” Pl"S 645% of fixed monthly charge, independent of the kWh (kilowatt- excess "V" $3M“) hours) used during the month. (A) Write a piecewise definition of the momth charge (A) Write a piecewise definition for the tax due T(x) on ' ' f d ll . S(x) for a customer who uses x kWh in a summer an income 0 x 0 ms month. (13) Graph T(x). B G h S . (C) Find the tax due on a taxable income of $20,000. Of ( ) mp (X) $35,000. (D) Would it be better for a married couple in Kansas with two equal incomes to file jointly or sepa- rately? Discuss. TABLE 3’; Summer (July—October) Base charge. $8.50 First 700 kWh or less at 0.0650/kWh Over 700 kWh at 0.090(1/k Wh 69. Physiology. A good approximation of the normal weight of a person 60 inches or taller but not taller than 80 inches is given by w(x) = 5.5): — 220, where x is height in inches C 1 i - t i u 66. Electricity rates. Table 4 shows the electricity rates charged 'md w( Y) 15 weigh m Pounds by Monroe Utilities in the winter months. (A) Write a piecewise definition of the monthly charge W(x) for a customer who uses x kWh in a winter month. TABLE '40 Winter (November—J Line) Base charge. $8.50 First 700 kWh or less at 0.0650/kWh Over 700 lel at 0.0530/kWh (B) Graph W(x). 67. State income tax. Table 5 shows a recent state income tax (A) D?5°Tibe how the graph 0f function w (fan be 0.13' schedule for married couples filing a joint return in the tamed from the graph of one of the basw functions state of Kansas. in Figure 1, page 65. ‘ (A) Write a piecewise definition for the tax due T(x) on (B) Skewh 2‘ graph 0f funaion w “Sing Pa“ (A) as an aid- { an income of x dollars. 70. Physiology. The average weight of a particular species of l‘ (B) Graph TC!) snake is given by -w(x) = 463x3. 0.2 s x s 0.8, where x ‘v l ' 1 th ' t d ' ' ht ' ‘ (C) Find the tax due on a taxable income of $40,000. or ‘5 mg "‘ me e” a" "’(x) '5 “mg ‘" gram $70,000. (A) Describe how the graph of function to can be ob- tained from the graph of one of the basic functions as State Income Tale in Figure 1, Page 65- SCHEDULE *MARRIED FILING JOINT (B) Sketch a graph of function '10 using part (A) as an aid. If taxable income is 71. Safety research. Under ideal conditions, if a person driving a vehicle slams on the brakes and skids to a stop, the speed ticular vehicle, 11(x) = 7.08\/J_t and 4 s x S 144. Over But Not Over Tax Due ls . . ‘ _ ' . $0 $3M“) 3.50% of taxable income of the vehicle 11(x) (1n miles per hour) is given approxl— .‘ $30100 $60000 $1.050 plus 6.25% of "lately .by Pm = CV; Where x 15 the length of Sk’d ' excess over $301“) marks (in teet) and C is a constant that depends on the i $6090” $2925 plus 6.45% of road conditions and the weight of the vehicle. For a par- 1 excess over $60,000 (A) Describe how the graph of function '1) can be ob- 68. State income tax. Table 6 shows a recent state income tax Pl“?! fmm ihe graph Of one Of the baSiC funcmms i schedule for individuals filing a return in the state of m Figure 1, page 65- . Kansas. . V (B) Sketch a graph of function '0 using part (A) as an aid. l 76 C H A P T E R 2 Functions and Graphs 72. Learning. A production analyst has found that on the av- erage it takes a new person T(x) minutes to perform a par- ticular assembly operation after x performances of the operation, where T(x) = 10 — W, 0 S x S 125. Section 2-3 FIGURE 1 Square function h(x) = x2 Explore 8: Discuss 1 (A) Describe how the graph of function Tcan be ob- tained from the graph of one of the basic functions in Figure 1. page 65. (B) Sketch a graph of function Tusing part (A) as an aid QUADRATIC FUNCTIONS I Quadratic Functions, Equations, and‘ Inequalities I Properties of Quadratic Functions and Their Graphs I Applications I More General Functions: Polynomial and Rational Functions If the degree of a linear function is increased by one, we obtain a second—degree func tion, usually called a quadratic function, another basic function that we will need ii our library of elementary functions. We will investigate relationships between quadratic functions and the solutions to quadratic equations and inequalities. Othe important properties of quadratic functions will also be investigated, includin; maximum and minimum properties. We will then be in a position to solve importan practical problems such as finding production levels that will produce maximum revenue or maximum profit. I Quadratic Functions, Equations, and Inequalities The graph of the square function h(x) = x2 is shown in Figure 1. Notice that th: graph is symmetric with respect to the y axis and that (0, 0) is the lowest point or the graph. Let‘s explore the effect of applying a sequence of basic transformations t! the graph of h. Indicate how the graph of each function is related to the graph of the functior h(x) = x2. Find the highest or lowest point, whichever exists, on each graph. (A) f(x)=(x—3)2—7=x2—6x+2 (B) g(x) = 0.5(x + 2)2 + 3 = 0.5;:2 + 2x + 5 (C) m(x) = —(x -— 4)2 + 8*: ~x2 + 8x — 8 (D) n(x) = —3(x + 1)2 — 1 = -3x2 — 6x — 4 Graphing the functions in Explore—Discuss 1 produces figures similar in shape It the graph of the square function in Figure 1. These figures are called parabolas. Thi functions that produced these parabolas are examples of the important class 0 quadratic functions, which we now define. DEFINITION Quadratic Functions If a, b: and c are real numbers with a at 0, then'the functior f(x). = axzk+ibx + .c ‘ Standard form' ' ‘ is a quadratic function and its graph is a parabola. V I If x is any real number, then ax2 + bx + c is also a real number. According to the agree- ment on domain and range in Section 2-1, the domain of a quadratic function is R, the set of real numbers. We will discuss methods for determining the range of a quadratic function later it ' this sectionflypical graphs of quadratic functions are illustrated in Figure 2. 90 C H A PT E R 2 Functions and Graphs (D) x = 2.490 or 12.530 million cameras (E) Losszl S x < 2.490 or 12,530 < x S 15; profit: 2.490 < x < 12.530 5. 22.9 mph =ax2+bx+c a='2.883?222'6 b=.5253543961 0= ‘43 . 13267968 Exercise 2-3 A In Problems 1—4, complete the square and find the standard form of each quadratic function. 1. f(x)=x2-4x+3 3. m(x)=—x2+6x—4 2.g(x)=x2-2x—5 4. n(x) = —x2 + 8x — 9 In Problems 5—8, write a brief verbal description of the rela- tionship between the graph of the indicated function (from Problems [—4) and the graph of y = x2. 5. f(x)=x2—4x+3 6. g(x)=x2—2x-5 L 7. m(x)=—x2+6x—4 8.n(x)=—x2+8x-9§ 9. Match each equation with a graph of one of the func- tions f, g, m, or n in the figure. For the functions indicated in Problems 11—14, find each of the following to the nearest integer by referring to the graphs for Problems 9 and 10. (A) Intercepts (B) Vertex (D) Range (F) Decreasing interval (C) Maximum or minimum (E) Increasing interval 11. Function n in the figure for Problem 9 12. Function m in the figure for Problem 10 13. Function fin the figure for Problem 9 14. Function g in the figure for Problem 10 In Problems 15—18, find each of the following: 5; (A) y = ‘(x + 2)2 + 1 (B) y = (x _ 2)2 _ 1 (A) Intercepts (B) Vertex (C) y = (x + 2)2 " 1 (D) y = ‘(x “ 2)2 + 1 (C) Maximum or minimum (D) Range ‘ 15. f(x) = —(x — 3)2 + 2 r y g f 16. g(x) = —(x + 2)2 + 3 r_J_ 3 17. m(x)=(x+l)2-2 ‘ "1 i 18. n(x)=(x—4)2—3 : r __ T. r i _ r J x B In Problems 19—22, write an equation for each graph in the — form y = a(x - h)2 + k, where a is either 1 or —1 and h and L‘” W]: ‘ kare integers L j J 19. 20. "I I1 Figure for 9 l l 10. Match each equation with a graph of one of the func- tions f, g, m, or n in the figure. ; (A)y=(x_3)2‘4 (C)y=—(x-3)2+4 m)y=—u+3Y+4 (D)y=(x+3)2—4 Figure for 10 In Problems 23—28, find the vertex form for each quadratic ftmction. Then find each of the following: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 23. f(x) = x2 — 8x + 12 24. g(x)=x2—6x+5 25. r(x) = --4x2 + 16x - 15 26. s(x) = ‘4x2 - 8x — 3 27. u(x) = 0.51:2 — 2x + 5 23. v(x) = 0.51:2 + 4x + 10 29. Let f (x) = 0.3):2 - x — 8. Solve each equation graphi- cally to four decimal places. (A) f(x) = 4 (B) f(x) = *1 (C) f(x) = -9 30. Let g(x) = -0.6x2 + 9x + 4. Solve each equation graphically to four decimal places. (A) 80‘) = *2 (B) 80‘) = 5 (C) 80‘) = 8 31. Let f(x) = 125x — 6x2. Find the maximum value off to four decimal places graphically. 32. Let f(x) = 100x — 7x2 — 10. Find the maximum value off to four decimal places graphically. Explain under what graphical conditions a quadratic function has exactly one real zero. Explain under what graphical conditions a quadratic function has no real zeros In Problems 35—38, first write each function in vertex form; then find each of the following (to four decimal plaCes): (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 35. g(x) = 0.25):2 — 1.5x — 7 36. m(x) = 0.20152 — 1.6x — 1 37. f(x) = ~o.12x2 + 0.96;: + 1.2 38. n(x) = ~0.15x2 — 0.90;: + 3.3 Tire mileage. An automobile tire manufacturer collected the data in the table relating tire pressure x (in pounds per square inch) and mileage (in thousands of miles): x Mileage 28 ' ' ‘45’ 30 52 32 55 34 51 36 47 ’£,~,.-'-~u a: “i 1? 4.2:. 1.30:, .5.-.. Quadratic Functions 91 - "p Solve Problems 39-44 graphically to two decimal places using a graphing calculator: 39. 2 — 5x - x2 = 40. 7+3x-2x2 =0 41. 1.91:2 — 1.5x — 5.6 < 0 42. 3.4 + 2.9): — 1.1;:2 2 0 43. 2.8 + 3.1): — 0.9x? s 0 44. 1.81:2 ~ 3.1x — 4.9 > 0 45. Given that f is a quadratic function with minimum f (x) = f (2) = 4, find the axis, vertex, range, and x intercepts 46. Given that f is a quadratic function with maximum f (x) = f(~3) = —5, find the axis, vertex, range, andx intercepts In Problems 47—50, (A) Graph f and g in the same coordinate system. (B) Solve f (x) = g(x) algebraically to two decimal places. (C) Solve f(x) > g(x) using parts (A) and (B). (D) Solve f(x) < g(x) using parts (A) and (B). 47. f(x) = -—O.4x(x — 10) g(x) = 0.3x + 5 0 s x s 10 48. f(x) = —O.7x(x - 7) g(x) = 0.5x + 35 05x57 49. f(x) = ~0.9x2 + 7.2x g(x) = 1.2): + 5.5 0 _<. x s 8 50. f(x) = -0.7x2 + 6.3x g(x) = 1.1x + 4.8 05x59 51. Give a simple example of a quadratic function that has no real zeros. Explain how its graph is related to the x axis 52. Give a simple example of a quadratic function that has exactly one real zero. Explain how its graph is related to the x axis A mathematical model for the data is given by f(x) = «0.518% + 33.31: — 481 (A) Complete the following table. Round values of f(x) to one decimal place. x Mileage fix) 28 45 3t) 52 32 55 34 5 l 36 47 92 C H A P T E R 2 Functions and Graphs (B) Sketch the graph of f and the mileage data in the same 57. Revenue. The marketing research department for a company ,A coordinate system. that manufactures and sells memory chips for microcomputers 3, (C) Use values of the modeling function rounded to two deci- established the following price—demand and revenue functions: mal places to estimate the mileage for a tire pressure of 31 . I 3 pounds per square inch. For 35 pounds per square inch. : 90‘) = 75 _ 3x Fr'ce'demand funct'm f: (D) Write a brief description of the relationship between tire R(") = xplx) = M75 “ 3X) Revenue 0100130" pressure and mileage. where p(x) is the wholesale price in dollars at which x million chips can be sold, and R(x) is in millions of dollars Both func- 54. Automobile production The table shows the retail market share I tions have domain 1 S x .<_ 20. of passenger cars from Ford Motor Company as a percentage 0f the US market‘ (A) Sketch a graph of the revenue function in a rectangular Yen Maflet Share coordinate system. W (B) Find the output that will produce the maximum revenue. 1980 172% What is the maximum revenue? 1985 ' v 13.8% (C) What is the wholesale price per chip (to the nearest , 1990 ' 2.00% dollar) that produces the maximum revenue? f 1995 7 20.7% E 1 2000 ' 20.2% , g . l i 2005' . . . ‘ 17.4% i . A mathematical model for this data is given by f(x) = —0.0206x2 + 0.548x + 16.9 where x = 0 corresponds to 1980. 58. Revenue. The marketing research department for a company that manufactures and sells “notebook” computers established 5" (A) Comple‘e the fouowmg tabla : the following price—demand and revenue functions: x M'fl‘e‘ Sh” fl” p(x) = 2,000 - 60x Price—demand function 0 I 172' ‘ ' R(x) = xp(x) Revenue function - 5 . 18‘s , = x(2,000 — 60x) .; 10 20.0 ' 15 " ' ’ 20.7 i, where p(x) is the wholesale price in dollars at which x thousand .20: 20a." [25 y ' 174‘ computers can be sold, and R(x) is in thousands of dollars Both functions have domain 1 s x s 25. are Aggro‘ikfltr‘iak aim, J (A) Sketch a graph of the revenue function in a rectangular coordinate system. (B) Find the output (to the nearest hundred computers) that will produce the maximum revenue. What is the maximum revenue to the nearest thousand dollars? (B) Sketch the graph of f and the market share data in the same coordinate system. (C) Use values of the modeling function f to estimate Ford’s market share in 2010. In 2015. (D) Write a brief verbal description of Ford’s market share from 1980 to 2005. (C) What is the wholesale price per computer (to the nearest dollar) that produces the maximum revenue? 2m.» an. earmammia. ;. Mayer. 59. Break-even analysis Use the revenue function from Problem 57 in this exercise and the given cost function: R(x) = x(75 — 3x) Revenue function C(x) = 125 + 16x C051: function where x is in millions of chips, and R(x) and C(x) are in mil- lions of dollars Both functions have domain 1 S x S 20. 55. Tire mileage. Using quadratic regression on a graphing calcu— @ lator, show that the quadratic function that best fits the data on . tire mileage in Problem 53 is ‘ f(x) = -0.518x2 + 33.3x ~ 481 l l i i l (A) Sketch a graph of both functions in the same rectangular i l i (COCffiCientS Tom‘ded to three Significant digits) 1 60. Break-even analysis Use the revenue function from Problem j l i coordinate system. I (B) Find the break-even points to the nearest thousand chips. T- (C) For what outputs will a loss occur? A profit? 56. Automobile production. Using quadratic regression on a graph- . % 58’ “1 “us exerme and the give" co“ funCt'On: ing calculator, show that the quadratic function that best fits R(x) = x(2,000 —— 60x) Revenue function the data on market share in Problem 54 is C(x) = 4,000 + 500x Cost function _ _ 2 for) _ 01206)! + 0548): + 16'9 where x is thousands of computers, and C(x) and R(x) are in thousands of dollars. Both functions have domain 1 s x s 25. (coefficients rounded to three significant digits). (A) Sketch a graph of both functions in the same rectangular coordinate system. (B) Find the break-even points, (C) For what outputs will a loss occur? Will a profit occur? Profit—loss analysis. Use the revenue and cost functions from Problem 59 in this exercise: R(x) = .r(75 - 3x) Revenue function C0) = 125 + [6.1: Cost function where x is in millions of chips. and R(x) and C(x) are in mil- lions of dollars. Both functions have domain 1 S. x s 20. (A) Form a profit function P, and graph R, C. and P in the same rectangular coordinate system. (it i Discuss the relationship between the intersection points of the graphs of R and C and the .r intercepts of P. (C) Find the x intercepts of P to the nearest thousand chips. Find the break—even points to the nearest thousand chips. (b) Refer to the graph drawn in part (A). Does the maxi— mum profit appear to occur at the same output level as the maximum revenue? Are the maximum profit and the maximum revenue equal? Explain. (E) Verify your conclusion in part (D) by finding the output (to the nearest thousand chips) that produces the maxi- mum profit. Find the maximum profit (to the nearest thousand dollars), and compare with Problem 578. ProfiIv/oss analysis. Use the revenue and cost functions from Problem 60 in this exercise: R(x) = x(2.000 - 60x) C(x) = 4,000 + 500x Kevcnue function Coat function where x is thousands of computers. and R(x) and C(x) are in thousands of dollars Both functions have domain 1 s x s 25. (A) Form a profit function P. and graph R, C, and P in the same rectangular coordinate system. (B) Discuss the relationship between the intersection points of the graphs of R and C and the x intercepts of P. (C) Find the x intercepts of P to the nearest hundred com- puters Find the break-even points. i D) Refer to the graph drawn in part (A). Does the maxi- mum profit appear to occur at the same output level as the maximum revenue? Are the maximum profit and the maximum revenue equal? Explain. 63. 64. 65. if} 66. Exponential Functions 93 (E) Verify your conclusion in part (D) by finding the output that produces the maximum profit. Find the maximum profit and compare with Problem 58B. Medicine. The French physician Poiseuille was the first to dis- cover that blood flows faster near the center of an artery than near the edge. Experimental evidence has shown that the rate of flow '1) (in centimeters per second) at a point x centimeters from the center of an artery (see the figure) is given by v = = Loom-0.04 —- x2) 0 g x 5 02 Find the distance from the center that the rate of flow is ‘20 centimeters per second. Round answer to two decimal places. Figure for 63 and 64 Medicine. Refer to Problem 63. Find the distance from the cen- ter that the rate of flow is 30 centimeters per second. Round answer to two decimal places. The table gives performance data for a boat powered by an Evinrude outboard motor. Find a quadratic regression model (y cut2 + bx + c) for boat speed y (in miles per hour) as a function of engine speed (in revolutions per minute). Estimate the boat speed at 3.100 revolutions per minute. Table for 65 and 66 rpm mph mpg [.500 4.5 8.2 2.000 5.7 6.9 2.500 7.8 4.8 3,000 9.6 4.l 3,500 13.4 3.7 The table gives performance data for a boat powered by an Evinrude outboard motor. Find a quadratic regression model (y = ax: + bx + c) for fuel consumption y (in miles per gal- lon) as a function of engine speed (in revolutions per minute). Estimate the fuel consumption at 2,300 revolutions per minute. Section 2-4 EXPONENTIAL FUNCTIONS I Exponential Functions in Base e Exponential Functions 3 Growth and Decay Applications I Compound Interest This section introduces the important class of functions called exponential fimcrions. These functions are used extensively in modeling and solving a wide variety of real- worid problems, including growth of money at compound interest; growth of populations of people, animals, and bacteria; radioactive decay: and learning associated with the mas- tery of such devices as a new computer or an assembly process in a manufacturing plant. CH‘Ap-rga 2VREV|EW_ ¥ whatnot.- TermS. Symbols. 80d Concepts: 1’11. ' Chapter 2 Review 11'. L 2.1 Functions 0 Point-by-point plotting may be used to sketch the graph of an equation in two variables: Plot enough ‘ points from its solution set in a rectangular cordinate system so that the total graph is apparent and then connect these points with a smooth curve. there corresponds one and only one element in the second sct.The first set is called the domain and the set of corresponding elements in the second set is called the range. 0 If x is a placeholder for the elements in the domain of a function, then x is called the independent variable or the input. if y is a placeholder for the elements in the range, then y is called the dependent variable or the output. 0 If in an equation in two variables we get exactly one ouput for each input, then the equation specifies a function.The graph of such a function is just the graph of the specifying equation. If we get more than one output for a given input, then the equation does not specify a function. 0 The Vertical-line test can be used to determine whether or not an equation in two variables specifies a function (Theorem 1, p. 51). I The functions specified by equations of the form y = mx + b, where m at 0, are called linear func- tions. Functions specified by equations of the form y = b are called constant functions. * 0 If a function is specified by an equation and the domain is not indicated, we agree to assume that the domain is the set of all inputs that produce outputs that are real numbers. ' ' The symbol ftx) represents the element in the range of f that corresponds to the element x of the domain. ’ ' Break-even and profit—loss analysis use a cost function C and a revenue function R to determine when a company will have a loss (R < C), will break even (R = C), or will have a profit (R > C). Typical cost, revenue, profit, and price-demand functions are given on page 55. 2'2 Elementary Functions: Graphs and Transformations 0 The graphs of six basic elementary functions (the identity function, the square and cube functions, the square root and cube root functions. and the absolute value function) are shown on page 65. 0 Performing an operation on a function produces a transformation of the l graph of the function. The basic graph transformations, vertical and horizontal translations (shifts), reflection in the x axis, and vertical stretches and shrinks, are summarized on page 69. 0 A piecewise-defined function is a function whose definition involves more than one formula. 2-3 Quadratic Functions . ' if a, b, and c are real numbers with a at 0, then the function f (x) = ax2 + bx + c Standard form is a quadratic function in standard form and its graph is a parabola. ' The quadratic formula ~bd: Vb2‘4ac 2a x= $«4ac20 can be used to find the x intercepts of a quadratic function. ‘ Completing the square in the standard form of a quadratic function produces the vertex form f(x) = a(x ~ h)2 + k vertex form ' From the vertex form of a quadratic function, we can read off the vertex, axis of symmetry, maximum or minimum, and range, and can easily sketch the graph (page 81). ' If a revenue function R(x) and a cost function C(x) intersect at a point (x0, m), then bOth this point and its x coordinate x0 are referred to as break-eveh points. - A function is a correspondence between two sets of elements such that to each element in the first set , Examples Ex. 1, p. 47 Ex. 2,p. 50 mm \1 cu: me» s. » mtTl xx mm mu: on» AM u w 'P'P'F’ '9'.“ tn :4 u: as Ex. 1, p. 64 Ex. 2, p. 66 Ex. 3, p. 67 Ex. 4, p. 68 Ex. 5, p. 69 Ex. 6, p. 70 Ex. 1, p. 77 Ex. 2, p. 81 Ex. 3, p.83 Ex. 4, p. 84 120 C H A P T E R 2 Functions and Graphs I Quadratic regression on a graphing calculator produces the function of the form y = air2 + bx + c that best fits a data set. D A quadratic function is a special case of a polynomial function, that is. a function that can be written in the form f(x) = aux" + anqxn‘1 + + tax + do for n a nonnegative integer called the degree of the polynomial. The coefficients a0, a1, . . . ,a,. are real numbers with an at 0. The domain of a polynomial function is the set of all real numbers. Graphs of representative polynomial functions are shown on page 87 and inside the front cover. 0 The graph of a polynomial function of degree n can intersect the x axis at most n times I The graph of a polynomial function has no sharp corners and is continuous, that is, it has no holes or breaks 0 A rational function is any function that can be written in the form f(x) = 32-3 where n(x) and d(x) are polynomials The domain is the set of all real numbers such that d(x) 7% 0. Graphs of representative rational functions are shown on page 88 and inside the front cover. d(x) at 0 0 Unlike polynomial functions, a rational function can have vertical asymptotes [but not more than the degree of the denominator d(x)] and at most one horizontal asymptote. 2-4 Exponential Functions 0 An exponential function is a function of the form f (x) = b’ where b at 1 is a pesitive constant called the base. The domain of f is the set of all real numbers and the range is the set of positive real numbers. 1 0 The graph of an exponential function is continuous, passes through (0, 1), and has the x axis as a horizontal asymptote. ->' 1, then [7" increases as x increases; if 0 < b < 1, then b" decreases as x increases (Theorem 1 , p. 95). 0 Exponential functions obey the familiar laws of exponents and satisfy additional properties (Theorem 2, p. 96). ‘ ' The base that is used most frequently in mathematics is the irrational number e z 2.7183. 0 Exponential functions can be used to model population growth and radioactive decay. 0 Exponential regression on a graphing calculator produces the function of the form y = abJr that beSt fits a data set. 0 Exponential functions are used in computations of compound interest: In! A = P(_1 Compound interest: formula (see summary on page 102). 2-5 Logarithmic Functions 0 A function is said to be one-to-one if each range value corresponds to exactly one domain value. ‘ The inverse of a one-to-one function f is the function formed by interchanging the independent and dependent variables off. That is, (a, b) is a point on the graph of f if and only if (b, a) is a point on the graph of the inverse of f. Apfunction that is not one-to—one does not have an inverse. 0 The inverse of the exponential function with base b is called the logarithmic function with base b, denoted y = log,, I. The domain of logbx is the set of all positive real numbers (which is the range of b’), and the range of logbx is the set of all real numbers (which is the domain of b"). ' Because 10gb x is the inverse of the function i)", Logarithmic form y = 10ng Exponential form is equivalent to x = by Ex. 5, p. 86 Ex. 1, p. 95 Ex. 2, p. 98 Ex. 3, p. 98 Ex. 4, p. 100 ‘ Ex.5,p.101 tions (Theorem l. p. 109). , Logarithm's to the base 10 are called common logarithms. often denoted simply by log x. Logarithms to the base 6 are called natural logarithms. often denoted by In x. o Logarithms can be used to find an investment’s doubling time—the length of time it takes for the value of an investment to double. 0 Logarithmic regression on a graphing calculator produces the function of the form y = a + b lnx that best fits a data set, REVIEW EXERCISE Work through all the problems in this chapter review and check your answers in the back of the book. Answers to all re— view problems are there along with section numbers in italics to indicate where each type of problem is discussed. Where weaknesses show up, review appropriate sections in the text. A In Problems 1—3, use point-by-point plotting to sketch the graph of each equation. 1. y = 5 — x2 2. x2 = y2 3. y2 = 4x2 4. Indicate whether each graph specifies a function: (A) y (B) (D) 5. For f(x) = 2x — 1 and g(x) = x2 — 2x, find: (A) f(-2) + g(-1) (B) f(0)-g(4) 8(2) f(3) (C) R55 (D) R5)" Properties of logarithmic functions can be obtained from corresponding properties of expontial func— Chapter 2 Review 121 Ex.4.p.l It) Ex.5.p4 lll) Ex.b,p. lll Ex.7,p. lll Ex.8,p. 1l2 Ex.9,p. 112 Ex. ll). p.114 Ex.ll,p.ll5 6. Write in logarithmic form using base e: u = e". 7. Write in logarithmic form using base 10: x = 10’. 8. Write in exponential form using base e: In M = N. 9. Write in exponential form using base 10: log it = v. Simplify Problems 10 and 11. 5x+4 en ti 1.. 11. (n) e—ll Solve Problems [2-14 for x exactly without using a calculator: 12. log x = 2 13. log: 36 = 2 14. log216 = x Solve Problems 15—18 for x to three decimal places 15. 10" = 143.7 16. e" = 503,000 17. log 1: = 3.105 18. In x = -1.147 19. Use the graph of function f in the figure to determine (to the nearest integer) x or y as indicated. (A) y = N» (B) 4 = for) (C) y = f(3) (D) 3 = f(x) (E) y = f(‘-6) (F) ‘1 = f(x) Figure for 19 122 C H A P T E R 2 Functions and Graphs 20. Sketch a graph of each of the functions in parts (A)—(D) using the graph of function f in the figure below. (A) y = —f(x) (B) y=f(x) +4 (C)y=f(x—2) (D)y=-f(x+3)-3 Figure for 20 21. Complete the square and find the standard form for the quadratic function f(x) = —x2 + 4x Then write a brief verbal description of the relationship between the graph off and the graph of y = x2. 22. Match each equation with a graph of one of the functions f,g, m, or n in the figure. Figure for 22 m9y=a—2r—4 m)y=-a—2r+4 (B)y=-—(x+2)2+4 (D)y=(x+2)2—4 23. Referring to the graph of function fin the figure for Prob- lem 11 and using known properties of quadratic functions, find each of the following to the nearest integer: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range (E) Increasing interval (F) Decreasing interval 24. Consider the set S of people living in the town of Newville. Which of the following correspondences specify a func- tion? Explain. (A) Each person in Newville (input) is paired with his or her mother (output). (B) Each person in Newville (input) is paired with his or her children (output). 25. The three points in the table are on the graph of the indi— cated function f. Do these three points provide sufficient in- l l l l i l i l i l l 3 g i l l l l l l l l formation for you to sketch the graph of y = f(x)? Add 3, more points to the table until you are satisfied that your . sketch is a good representation of the graph of y = f(x) on the interval [—5, 5]. In Problems 26—29, each equation specifies a function. Deter- mine whether the function is linear, quadratic, constant, or none of these. 2.6.)»=4--Jc+3x2 27.y=1+65x 28.y=7;x4x 29.y=8x+2(10-4x) salve Fromm” 3037 for 7‘ exactly Without using a calculator. _ 30. log(x + 5) = log(2x — 3)31. 21n(x — 1) = ln(Jt2 - 5) 32. 9N = 31” 33. e21 = (3‘1" 34. 2x2e" = I’ixeJlr 35. logm 9 = x 36. logx8 = -3 3'7. loggx = % Solve Problems 38—4 7 for x to four decimal places. 38. x = 3(e1-49) 39. x = 2300041161) 40. logx = —2.0144 41. In x = 0.3618 42. 35 = 7(3") 43. 0.01 = (0'05" 44. 8,000 = 4,000(1.08") 45. 52M = 7.08 46. x = log27 47. x = logo-2 5.321 48. Find the domain of each function: 2x - 5 3x A = ———-——-—— B = <>na fi‘x_6 <>nn V€:; 49. The functiongis defined by g(x) = 2x — 3%. Translate I into a verbal definition. 50. Find the vertex form for f(x) = 4x2 + 4x — 3 and then. find‘the intercepts, the vertex, the maximum or minimum, ,, and the range. 51. Let f(x) = e" — 1 and g(x) = ln (3: + 2). Find all points of intersection for the graphs off and g. Round answers to‘ two decimal places In Problems 52 and 53, use point-by-point plotting to sketch the graph of each function. 50 52. f(x) — x2 + 1 —66 53. f(x) — 2 + x2 If f (x) = 5x + 1, find and simplify each of the following in Problems 54—5 7. 54‘ f (f (0)) 55- f (f (—1)) 56. f(2x — 1) s7. f(4 - x) 58. Let f(x) = 3 - 2):. Find (A) fl2) (B) f(2 + h) h _. (C) m + h) — f(2) (D) f_(2+_})l_£?l 59'. Let f(x) = x2 — 3x + 1. Find (A) fit!) f (f1 + h) ‘ f(tl) (C) f(tt + 11) —- —-T—— (:0. Explain how the graph of m(.r) = —Ix — 4| is related to the graph of y = lxl. (,1. Explain how the graph of g(x) = 0.33:3 + 3 is related to the graph of y = x3. (B) f(fl + h) 62. The following graph is the result of applying a sequence of ' transformations to the graph of y = x2. Describe the trans- formations verbally and write an equation for the graph. Figure for 62 V 63. The graph of a function f is formed by vertically stretching the graph of y = V? by a factor of 2, and shifting it to the left 3 units and down 1 unit. Find an equation for function fandgraph it for -5 S x S 5 and —5 S y S S. In Problems 64—71, discuss the validity of each statement. If the statement is always true, explain why. If not, give a counter- example. 64. Every polynomial function is a rational function. 65. Every rational function is a polynomial function. 66. The graph of every rational function has at least one verti- cal asymptote. 67. The graph of every exponential function has a horizontal asymptote. 68. The graph of every logarithmic function has a vertical as- ymptote. 69. There exists a logarithmic function that has both a vertical . and horizontal asymptote. 70. There exists a rational function that has both a vertical and horizontal asymptote. 71. There exists an exponential function that has both a verti- ' cal and horizontal asymptote. Graph Problems 72-74 over the indicated interval. Indicate increasing and decreasing intervals. 72. y = 2""; [—2, 4] 73. f(t) = 10e‘0‘03’;t 2 0 74. y = 1n(x + 1); (—1.10] 75. Sketch the graph of f for x 2 0. 9+0.3x ifOSxSZO “’0 “ {5 + 0.2x ifx > 20 76. Sketch the graph of g for x 2 0. 0.5x+5 ifOstlo g(x) = 1.2x —— 2 if10 < x S 30 2x - 26 ifx > 30 C 80. Noting thatrr = 3.141 592 654 . . ..and Chapter 2 Review 123 77. Write an equation for the graph shown in the form y = a(x ~ h)1 + k, where a is either -1 or +1 and h and k are integers. Figure for 77 78. Given f(x) = —0.4x2 + 3.21 + 1.2, find the following al- gebraically (to three decimal places) without referring to a graph: (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range 79. Graph f(x) = —0.4x2 + 3.2x + 1.2 in a graphing calcu— is: lator and find the following (to three decimal places) using TRACE and appropriate built-in commands: ' (A) Intercepts (B) Vertex (C) Maximum or minimum (D) Range V? = 1.414 213 562 . . . explain why the calculator results shown here are obvious. Discuss similar connections be- tween the natural logarithmic function and the exponential function with base 2. Solve Problems 81—84 exactly without using a calculator: 81. logx - log3 = log4 — log(x + 4) 82. ln(2x — 2) -— ln(x ~ 1) = lnx 83. ln(x + 3) - lnx = 21n2 84. log 3x2 = 2 + log 9x ? 85. Write In y = -St + 1n c in an exponential form free of log- arithms Then solve for y in terms of the remaining variables 86. Explain why 1 cannot be used as a logarithmic base. 87. The following graph is the result of applying a sequence of transformations to the graph of y = W. Describe the trans- formations verbally, and write an equation for the graph. \ lill L‘ I II... E!- -IIIIIII- Figure for 87 124 C H A P T E R 2 Functions and Graphs 88. Given C(x) = 0.3):2 + 1.2x —- 6.9, find the following alge— 1 89. Graph G(x) = 0.33:2 + 1.2x — 6.9 inastandard viewing braically (to three decimal places) without the use of a graph: Window. Then find each of the following (to three decimal ‘ (A) Intcmcpts j places) using appropriate commands. (B) Vertex (A) Intercepts (C) Maximum or minimum ‘ (B) VCYtCX (D) Range (C) Maximum or minimum (D) Range (E) Increasing and decreasing intervals (E) Increasing and decreasing intervals Applications In all problems involving days, a 365-day year is assumed. (B) Graph S (x ). 90. Egg consumption. The table shows the annual per capita con- Energy Charge (June-September) sumption of eggs in the United States. '~ - . 4 . u - v \ $3.001for‘the firstZQ kWhOr less. I I 5.7mm: kthor thence 180 WM 7 » . r Y N be ME - . ,~ » 1 ., 7e.“ .“m I t g 53.46; puma fortho sexism kWh- v 271: 2.1_7¢ pet kWh mart byertnoo kWh 93. Money growth. Provident Bank of Cincinnati, Ohio recently offered a certificate of deposit that paid 5.35% compounded quarterly. If a $5,000 CD earns this rate for 5 years, how much will it be worth? 94. Money growth. Capital One Bank of Glen Allen. Virginia recently offered a certificate of deposit that paid 4.82% com- - pounded daily. If a $5,000 CD earns this rate for 5 years, how much will it be worth? 95. Money growth. How long will it take for money invested at 6.59% compounded monthly to triple? (A) Complete the following table. 96. Money growth. How long will it take for money invested at k 7.39% compounded daily to double? 97. Break—even analysis. The research department in a company that manufactures AM/FM clock radios established the fol- lowing pficHemmd, cost, and revenue functions: A mathematical model for this data is given by f(x) = 0.181x2 — 4.88x + 271 where x = 0 corresponds to 1980. i i l l i l l i i i i I l l t l x Consumption fix) p(x) = 50 — 1.25x Price—demand function C(x) = 160 + 10x Coat function R0) = x1107) = x(50 — 1.25x) Kevenue function where x is in thousands of units, and C(x) and R(x) are in thousands of dollars. All three functions have domain 1 s x s 40. (A) Graph the cost function and the revenue function simu- taneously in the same coordinate system. (B) Determine algebraically when R = C.Then, with the aid j of part (A), determine when R < C and R > C to the 91. Egg consumption. Using quadratic regression on a graphing nears“ “nit- calculator, show that the quadratic function that best fits the ' (C) Determine algebraically the maximum revenue (to the data on egg consumption in Problem 90 is nearest thousand dollars) and the output (to the nearest f (x) = 0.181x2 - 4.881: + 271 unit) that produces the maximum revenue. What is the (coefficients rounded to three significant digits). “10135313 Price Of the radio (to the nearest (10113! ) at this 92. Electricity rates. The table shows the electricity rates charged ompm? by Easton Utilities in the summer months. (A) Write a piecewise definition of the monthly charge S(x) (in dollars) for a customer who uses x kWh in a summer month. (B) Graph y = f (x) and the data in the same coordinate system. (C) Use the modeling function f to estimate the per capita egg consumption in 2012. I (D) Based on the information in the table, write a brief de- scription of egg consumption from 1980 to 2005. 98. Profit-loss analysis Use the cost and revenue functions from Q Problem 97. (A) Write a profit function and graph it in a graphing calculator. . .t'mehzv‘ " (3) Determine graphically when P = 0, P < 0, and P > 0 to the nearest unit. (C) Determine graphically the maximum profit (to the near— est thousand dollars) and the output (to the nearest unit) that produces the maximum profit. What is the wholesale price of the radio (to the nearest dollar) at this output? [Compare with Problem 97C.] 99. Construction. A construction company has 840 feet of chain- link fence that is used to enclose storage areas for equipment and materials at construction sties. The supervisor wants to set up two identical rectangular storage areas sharing a common fence (see the figure). {- L._Ly_i Assuming that all fencing is used, (A) Express the total area A(x) enclosed by both pens as a function of x. (B) From physical considerations, what is the domain of the function A ? (C) Graph function A in a rectangular coordinate system. (D) Use the graph to discuss the number and approximate locations of values of x that would produce storage areas with a combined area of 25,000 square feet. (E) Approximate graphically (to the nearest foot) the val- ues Of): that would produce storage areas with a com- bined area of 25,000 square feet. (F) Determine algebraically the dimensions of the storage areas that have the maximum total combined area. What is the maximum area? Equilibrium point. A company is planning to introduce a 10-piece set of nonstick cookware. A marketing company established price—demand and price—supply tables for selected prices (Tables 1 and 2), where x is the number is willing to sell each month at a price of p dollars per set. TABLE ‘1 PHI: Utjrr‘~.ii:(i x p = D(x) 6) 985 330 2.145 225 2,950 170 4,225 105 5,100 50 TABLE 2 Price- Simply x P = 50*) (S) 985 30 2,145 75 2,950 110 4.225 155 5,100 gj 190~ of cookware sets people are willing to buy and the company Q 103. Chapter 2 Review 125 (A) Find a quadratic regression model for the data in Table 1. Estimate the demand at a price level of $180. (B) Find a linear regression model for the data in Table 2. Estimate the supply at a price level of $180. (C) Does a price level of $180 represent a stable condition, or is the price likely to increase or decrease? Explain. (D) Use the models in parts (A) and (B) to find the equilibrium point. Write the equilibrium price to the nearest cent and the equilibrium quantity to the nearest unit. 101. Telecommunications. According to the Telecommunications Industry Association, wireless telephone subscriptions grew from about 4 million in 1990 to over 180 million in 2005 (Table 3). Let x represent years since 1990. TABLE 3 Wu clots Telephone Sums! ~ mum Year Million Subscribers 1990 4 1993 13 1996 38 1999 76 2002 143 2005 180 (A) Find an exponential regression model (y = ab‘) for this data. Estimate (to the nearest million) the number of subscribers in 2015. (B) Some analysts estimate the number of wireless subscribers in 2018 to be 300 million. How does this compare with the prediction of the model of part (A)? Explain why the model will not give reliable predictions far into the future. 102. Medicine. One leukemic cell injected into a healthy mouse will divide into 2 cells in about % day. At the end of the day these 2 cells will divide into 4. This doubling continues until 1 billion cells are formed; then the animal dies with leukemic cells in every part of the body. (A) Write an equation that will give the number N of leukemic cells at the end of t days (B) When, to the nearest day, will the mouse die? Marine biology. The intensity of light entering water is reduced according to the exponential equation = Ine‘kd where I is the intensity d feet below the surface, 10 is the intensity at the surface, and k is the coefficient of extinction. Measurements in the Sargasso Sea in the West Indies have indicated that half of the surface light reaches a depth of 126 C H A PT E R 2 Functions and Graphs 73.6 feet. Find k (to five decimal places), and find the depth 105, Population growm Some countries have a relative growth (to the nearest foot) at which 1 % of the surface light rate of 3% (or more) per year. At this rate, how many years (to remains. ' ~ the nearest tenth of a year) will it take a population to double? 104. Agriculture. The number of dairy cows on farms in the United States is shown in Table 4 for selected years since ‘ 1950. Let 1940 be year 0. 106. Medicare. The annual expenditures for Medicare (in bil- lions of dollars) by the US. government for selected years since 1980 are shown in Table 5. Let x represent years since 1980. TABLE 4 ‘ ’1 H , ‘ I . TABLE 5 Main Lin” [vpv'mhtu'm Year Dairy Cows Year Billion 5 Mom") ' 1980 ' 37 1950, L Vs, v W “A. V ' 23.853; ‘ ' ‘ 1985 72 1960“ - ~: > 19527 ' 1990‘ . V I V1111 1970 y ‘ " 12.09" ' ' 1995 ' r 181 1930 r , q . 10.758 ' _ 2000 r I » V V': 1'97. 1990 ' ‘ 10,0157“ * r- 2005 . 3’36 20011;, '_j p 9.19.0 t (A) Find an exponential regression model (y = ab") for the (A) Find a logarithmic regression model (y = a + b ln x) f data. Estimate (to the nearest billion) the annual ex- for the data. Estimate (to the nearest thousand) the penditures in 2015. “unmet 0f dalry cows "1 2015' (B) When will the annual expenditures reach one trillion (B) Explain why it is not a good idea to let 1950 be year 0. dollars? ...
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MTH101_Chap2_4 - A: Functions 59 (D) 5(E) 200 g I [5 g Y .5...

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