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1 Answer Ke Masters

2 Copright The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced onl for classroom use; be provided to students, teacher, and families without charge; and be used solel in conjunction with Glencoe Advanced Mathematical Concepts. An other reproduction, for use or sale, is prohibited without prior written permission of the publisher. Send all inquiries to: Glencoe/McGraw-Hill 8787 rion Place Columbus, H ISBN: AMC Answer Ke Masters

3 Chapter Contents Page Linear Relations and Functions... Sstems of Linear Equations and Inequalities... The Nature of Graphs Polnomial and Rational Functions The Trigonometric Functions... 6 Graphs of the Trigonometric Functions Trigonometric Identities and Equations Vectors and Parametric Equations Polar Coordinates and Comple Numbers Conics... 6 Eponential and Logarithmic Functions... Sequences and Series... 9 Combinatorics and Probabilit Statistics and Data Analsis Introduction to Calculus iii

4 Chapter Linear Relations and Functions - Relations and Functions Pages 8.. Sample answer: Determine whether a vertical 4. Keisha is correct. Since a line can be drawn through the function can be epressed as a graph so that it passes through set of ordered pairs, a function more than one point on the is alwas a relation. However, graph. Since it does, the graph in a function, there is eactl does not represent a function. one -value for each -value. Not all relations have this constraint {(, 4), (0, 0), (, 4), (6, 8)}; D {, 0,, 6}; R {8, 4, 0, 4} Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

5 7. {(6, ), (4, 0), (, 4), 8. (, ), (4, )}; D {6, 4,,, 4}; R {4, 0,, } {, 0,, }; {6, 0,, 4}; es; each member of the domain is matched with eactl one 5 member of the range {,, 6}; {6,, 0, 4}; no; a. domain : all reals; range: all 6 is matched with two members reals of the range. b. Yes; the graph passes vertical line test m 5. 6a. {(8, 40), (8, 0), (8, 45), (78, 00), (8, 55), (7, 00), (80, 5), (77, 0), (78, 90), (7, 80), (86, 00), (77, 0), (8, 60)}; {78, 7, 77, 80, 8, 8, 8, 86}; {80, 90, 00, 0, 5, 0, 40, 45, 55, 60, 00} Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

6 6b. 6c. no {(5, 5), {, ), (, ), 4 4 (, )}; D {5,,, }; R {5,,, } Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

7 . {(0, 0), (5, 0), {0, 0), (5, 0)};. {(4, 0), (5, ), (8, 0), (, )}; D {0, 5, 0, 5}; R {0} D {4, 5, 8, }; R {0, }. {(, ), (, ), (0, 0), (, )}; 4. {(5, 5), (, ), (, ), D {,, 0, }; (, ), (4, 4)}; R {, 0, } D {5,,,, 4}; R {4,,,, 5 5. {(, 4), (, ), (, 0), (, ), 6. (, )}; D {}; R {4,, 0,, } Glencoe/McGraw-Hill 4 Advanced Mathematical Concepts Chapter

8 .. {4, 5, 6}; {4}; es; Each -value is paired with eactl one -value {}; {6,, 0, 4}; No; the 4. {0,, 4}; {,, 0,, }; No; -value is paired with more the -values and 4 are paired than one -value. with more than one -value. 5. {0,, 5}; {8,, 0,, 8}; No; 6. {., 0.4, 0.}; {, }; the -values and 5 are paired Yes; each -value is paired with with more than one -value. eactl one -value. 7. {9,, 8, 9}; {, 0, 8}; Yes; 8. Domain: all reals; Range: all each -value is paired with reals; not a function because eactl one -value. it fails the vertical line test. 9. Domain: {,,,,, }; 40. Domain: { 8 8}; Range: {,,, }; a function Range: { 8 8}; not a because each -value is paired function because it fails the with eactl one -value. vertical line test a 45. n 5n m or or 7 5a. 5b. 5 5c., 5a. {(944, 66), (8,584, 697), 5b. (8,40, 5805), (6,8, 999), (,589, 6), (8,9, 5950), (,877, 6)}; {944, 8,584, 8,40, 6,8,,589, 8,9,,877}; {66, 697, 5805, 999, 6, 5950, 6} b 6 b 7 6 Number 5 Attending (thousands) Number Applied (thousands) Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts Chapter

9 5c. Yes; no member of the domain is paired with more than one member of the range a. 9.5 F 54b. 9.5 F 54c. 90 F 54d. 70 F 54e. 55 F 55a.4,989,6.9 m; ,958, 49.6 m; 49,709,44. m;,768,775,50 m 55b.,98,96.64 m 57. B Composition of Functions Pages 7 9. Sample answer: f(). Iteration is composing a and g() 6; function on itself b evaluating Sample eplanation: the function for a value and Factor 6. then evaluating the function on that function value.. No. [f g]() is the function f() 4. Sample answer: Composition performed on g() and [g f ]() of functions is performing one is the function g() performed function after another. An on f(). See students everda eample is putting on countereamples. socks and then putting shoes on top of the socks. Buing an item on sale is an eample of when a composition of functions is used in a real-world situation ; 4; 6. ; ; 4 5 9, ; Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts Chapter

10 9. 5,, 5 0a. K(F) (F ) b..5, 48.7, 55.7, 7.5, ; 9;., ; 7 8;, ;, 9 9, ;, or. 5, 7; 4. 5, 5; 4 5, , 7; 5; 6, 5; 7 5 5, 7; 5, 0 or 5 7, 5, 0, or ; ; 7. 4; ; ; ; 7., ;, 0. all reals ,, 7 6., 5, 6 7.,, 8. Yes; the total with the discount and ta is $9.0. Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts Chapter

11 9. Yes; If f() and g() are both 0a. W n d(f p F f ) lines, the can be represented 0b. 000 J as f() m b and g() m b. Then [f g]() m (m b ) b m m m b b. Since m and m are constants, m m is a constant. Similarl, m, b, and b are constants, so m b b is a constant. Thus, [f g]() is a linear function if f() and g() are both linear. a. h[f ()], because ou must. subtract before figuring the bonus. b. $750 a. v(p) b. r(v) 0.84v 47p c. r(p) 75 d. $5.94, $8., $ {(, 8), (0, 4), (, 6), (5, 9)}; D {, 0,, 5}; R {9, 6, 4, 8} p 47 4a. (ear, interest): (, $400), (, $4), (, $466.56), (4, $50.88), (5, $544.0) 4b. {,,, 4, 5}; {$400, $4, $466.56, $50.88, $544.0} 4c. Yes; for each member of the domain there is eactl one corresponding member of the range. 6. D {,,, 4}; R {5, 6, 7, 8}; es, ever element in the domain is paired with eactl one element of the range C Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts Chapter

12 - Graphing Linear Equations Pages -5. m represents the slope of the. 7; the line intercepts the -ais graph and b represents the at (7, 0). -intercept.. Sample answer: Graph the 4. Sample answer: Both graphs -intercept at (0, ). Then move are lines. Both lines have a down 4 units and right unit to -intercept of 8. The graph of graph a second point. Draw a line to connect the points slopes upward as ou move from left to right on the graph and the graph of 5 8 slopes downward as ou move from left to right on the graph. ( ), 0 (0, ) ( 0, 5 ) (5, 0) (, 8) (0, 7) (0, 5) none Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts Chapter

13 a. (8.500, 7), (44.5, 88). b..667 c. For each -centimeter increase in the length of a man s tibia, there is a.667-centimeter increase in the man s height Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts Chapter

14 f() f() (, 9 0) 5. f() 6. f() f() 4 f() (, 0) (, ) none f() 8 4 f() (0, 0) 4 f() f() 8 4 Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

15 Sample answer: f () 5; f () 0 a. 0.4 ohm 4. 4 b..4 volts 5a. 6. No; the product of two positives 4 is positive, so for the product of 5b. For each -degree increase in the slopes to be, one of the the temperature, there is a slopes must be positive and the other must be negative. -pascal increase in the 4 5c. pressure. 7a. 6; the software has no 8. A function with a slope of 0 has monetar value after no zeros if its -intercept is not 6 months. 0, a function with a slope of 0 7b. 90; for ever -month has an infinite number of zeros change in the number of if its -intercept is 0, a function months, there is a $90 with an slope other than 0 decrease in the value of the has eactl zero. software. Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

16 7c. v(t) (0, 0,440) 0, (6, 0) t 9a , 4 9b. $55.0 9c d. $5.70 4a. d(p) 0.88p 4. 50, 4 4b. r (d) d 00 4c. r (p) 0.88p 00 4d. $60.99, $779.99, $ No; the graph fails the vertical line test. 45. {(, 4), (, ), (, ), 46. D (0, )}, es Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

17 -4 Writing Linear Equations Pages 9. slope and -intercept; slope. Sample answer: and an point; two points Use point-slope form. m( ) (4) ( ) Use slope-intercept form. m b 4 () b b Substitute the slope and intercept into the general form Write in standard form represents the hourl rate 4. and 49 represents the fee for coming to the house. 5. Sample answer: When given the slope and the -intercept, use slope-intercept form. When given the slope and a point, use point-slope form. When given two points, find the slope then use point-slope form a b. 4. in. 0c. Sample answer: No; the grass could not support its own weight if it grew that tall. Glencoe/McGraw-Hill 4 Advanced Mathematical Concepts Chapter

18 A 5a. t 6. m 000 B 5b. about 5.7 weeks 7a. Sample answer: Using (0, 8) 8a. See students work. 9 6 and (7, 7),. 8b. Sample answer: nl two 7 7 points were used to make the 7b. using sample answer from prediction equation, so man part a, 6.7 mpg points lie off of the line. 7c. Sample answer: The estimate is close but not eact since onl two points were used to write the equation. 9. Yes; the slope of the line through 0. 9 (5, 9) and (, ) is or The slope of the line through (, ), and (, 6) is or 4 6 (). Since these two lines would have the same slope and would share a point, their equations would be the same. Thus, the are the same line and all three points are collinear. a. $6 billion. 4 b. The rate is the slope , 4. {(4, 6), (, 9), (, 4)}, es 7 5. A Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts Chapter

19 Chapter Mid-Chapter Quiz Page. {,, 4}, {8,,, 7};. 9 No, in the domain is paired with more than one element of the range.. n 4. $65 5., 0;, a. about 9,975 0b. 9,975 7,04,84 Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts Chapter

20 -5 Writing Equations of Parallel and Perpendicular Lines Pages 5 7. If A, B, and C are the same or. The have no slope. the ratios of the As and the Bs and the Cs are proportional, then the lines coincide. If A and B are the same and C is different, or the ratios of the As and the Bs proportional, but the ratio of the Cs is not, then the lines are parallel. 4. ; 4. All vertical lines have undefined 4 slope and onl horizontal lines are perpendicular to them. The slope of a horizontal line is none of these 6. perpendicular 7. parallel 8. coinciding parallelogram. perpendicular. parallel 4. none of these 5. perpendicular 6. coinciding 7. perpendicular 8. none of these 9. coinciding 0. parallel. None of these; the slopes are. 8 0 neither the same nor opposite reciprocals a b a. 4 0a. Sample answer: 0, b. 0b. Sample answer: 7 0, Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts Chapter

21 ; 7;. We are given m b and m b with m m and b b. Assume that the lines intersect at point (, ). Then m b and m b. Substitute m b for in m b. Then m b m b. Since m m, substitute m for m. The result is m b m b.subtract m from each side to find b b.however, this contradicts the given information that b b. Thus, the assumption is incorrect and the lines do not share an points. a. No; the lines that represent the 4a ; situation do not coincide ; b. Yes; the lines that represent ; the situation coincide b. parallel lines; no 4c. 05.6, 45.78, 5.0, 06.; No; the equations take onl one pair of das into account a b. $6.75 6c. $ Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts Chapter

22 9. Sample answer: {(, 4), (, 4), (, ), (, ), (0, 0)}; because the -values and are paired with more than one -value. -6 Modeling Real-World Data with Linear Functions Pages the rate of change. Choose two ordered pairs of data and find the equation of the line that contains their graphs. Find a median-fit line b separating the data into three sets and using the medians to locate a line. Use a graphing calculator to find a regression equation.. Sample answer: age of a car 4a. and its value Personal Consumption on Durable Go Dollars Year b. Sample answer: Using (99, 800) and (994, 66), ,40. 4c ,89.4; r d. $76.60; Yes, the correlation value shows a strong correlation. Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts Chapter

23 5a. Computers in Schools 6a Average b. Sample answer: Using (9, 77) and (40, 4), Year c ; r b. Sample answer: Using 6d. 50; Yes, r is fairl close to. (987, ) and (996, 7.8), (Actual data is 54.) c. 6.8,50.4; r 0.8 5d. 995; No; in 995 there were 0 students per computer. 7a. Personal Income 8a Dollars (thousands) Year Car Weight and Mileage 7b. Sample answer: Using 8b. Sample answer: Using (99, 9,00) and (7.5, 65.4) and (5.0, 7.7), (995,,), ,087, c ; r c. 058.,076,9.64; 8d. 0.9; No, r doesn t show a r 0.99 particularl strong relationship. 7d. $,77.96; Yes, r shows a strong relationship. 400 Wins Average Mileage 40 0 All-Time NFL Coaching Victo Years Weight (hundreds of pou Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts Chapter

24 9a. 0a. 9b. Sample answer: Using (0., ) and (.4, 7900), 0b. Sample answer: Using (988, 5.) and (997, 0.8), 9c ; r d. The correlation value does not 0c ; show a strong or moderate r relationship. 0d. 8.4%; Yes, r is etremel close to. a. World Population a. Sample answer: the space 7000 shuttle; because anthing less 6000 than perfect could endanger 5000 the lives of the astronauts. Millions 4000 b. Sample answer: a medication of People 000 that proves to help dela the progress of a disease; because an positive 0 correlation is better than none Year or a negative correlation. c. Sample answer: comparing a b. Sample answer: Using (, 00) dosage of medicine to the and (998, 5900), growth factor of cancer cells; because the greater the c ; r 0.56 dosage the fewer cells that d. 979 million; No, the correlation are produced. value is not showing a ver strong relationship. Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

25 . The rate of growth, which is 4a. No; the lines do not coincide. the slope of the graphs of the 4b. Yes; the lines coincide. regression equations, for the women is less than that of the men s rate of growth. If that trend continues, the men s median salar will alwas be more than the women s a. 6b. $4 billion 6c. If the nation had no disposable income, personal consumption ependitures would be $4 billion. For each $ billion increase in disposable income, there is a 0.8 billion dollar increase in personal consumption ependitures. 7. ; 8. Yes; each domain value is paired with eactl one range value. 9. C Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

26 -7 Piecewise Functions Pages f () if 0. reals, even integers if 0 if 0. f () if Ale is correct because he is if 4 appling the definition of a function Greatest integer function; h is 0. shuttle facilit hours, c (h) is the cost, c (h) 50h if [[h]] h. 50[[h ]] if [[h]] h Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

27 Glencoe/McGraw-Hill 4 Advanced Mathematical Concepts Chapter

28 Step; t is the time in hours, c(t) 4. Greatest integer; w is the is the cost in dollars, weight in ounces, c(w) is the cost in dollars. d(t) if t 0 if t. c(t) 6 if t 4 if t (w ) if [[w]] w c(w) [[w]] if [[w]] w c(w) t 4 w Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts Chapter

29 5. Absolute value; w is the weight 6a. step in pounds, d(w) is the 6b. v is the value of the order, discrepanc, d(w) w. s(v) is the shipping,.50 if 0.00 v if 5.0 v s(v) if 75.0 v if 5.0 v 6c. 7. If n is an integer, then all 8a. absolute value ordered pairs (, ), where 8b. d(t) 65 t and are both in the interval 8c. [n, n ) are solutions. 8d. 9.5 heating degree das Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts Chapter

30 9a. step 0. No; The functions are the same 9b. 6% if $0,000 if is positive. However, if is t () 8% if $0,000 $0,000 negative, the functions ield 9.5% if $0,000 different values. For eample, 9c. [g f ](.5) and [f g](.5) ; [g f ](.5) and [f g](.5). 9d. 9.5% a b. Sample answer: Using (,8,088, 5.4) and (6,777,.), c , r 0.68 d. 8.7%; No, the actual value is %. Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts Chapter

31 a. (9, 9), (, 5) 4. p() b. c. The average number of points scored each minute. 5. $ {7,, 0, 4, 9}; {, 0,,, }; Yes; no value of the domain is paired with more than one member of the range. 7. A -8 Graphing Linear Inequalities Pages Graph the lines and 7. The graph of is solid and the graph of 7 is dashed. Test points to determine which region(s) should be shaded. Then shade the correct region(s).. Sample answer: The boundaries 4. separate the plane into regions. Each point in a region either does or does not satisf the inequalit. Using a test point allows ou to determine whether all of the points in a region satisf the inequalit. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts Chapter

32 a. c(m) m 8b. c(m) c(m) m m 8c. Sample answer: (0, 45), (0, 49), (0, 50) < > 5 Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts Chapter

33 Glencoe/McGraw-Hill 0 Advanced Mathematical Concepts Chapter

34 a b. c. Sample answer: (0, 48), (60, 0), (45, 6) d. Sample answer: Using comple computer programs and sstems of inequalities. Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

35 5a. points in the first and third 6a. 8 ; 500 quadrants 5b. If and satisf the inequalit, 4 4 then either 0 and 0 or 6b. 0 and 0. If 0 and 0, then and 4.5. Thus,. Since is positive, 4.5. If 0 and , then and. Then () or ( ). Since both and are negative, ( ) is negative, and ( ). 7a. 0.6(0 a) r 0.9(0 a) 8a. step 7b. r 00 8b. Let c(h) represent the cost for h hours. c(h) 55h if [[h]] h 0.6(0 a) r 0.9(0 55[[h ]] if [[h]] h 8c a Hours 9a b. 6 0 a. (0, ), (6, 48);.565. E b. the average change in the temperature per hour Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

36 Chapter Sstems f Linear Equations and Inequalities - Solving Sstems of Equations in Two Variables Pages Sample answer: 4 7 The substitution method is usuall easier to use whenever one or both of the equations are alread solved for one variable in terms of the other.. Sample answer: consistent sstems of equations have at least one solution. A consistent, independent sstem has eactl one solution; a consistent, dependent sstem has an infinite number of solutions. An inconsistent sstem has no solution. See students work for eamples and solutions. 5. (, ) 5. Sample answer: Madison might consider whether the large down-pament would strap her financiall; if she wants to bu the car at the end of the lease, then she might also consider which lease would offer the best buout. 4. Inconsistent; sample answer: the graphs of the equations are lines with slope, but each equation has a different -intercept. Therefore, the graphs of the two equations do not intersect and the sstem has no solution. 6. no solution (, ) (, 5) 9. (6, 4). consistent and independent 8., C 0.,000 baseball, 000 karate. inconsistent Glencoe/McGraw-Hill Advanced Mathematical Concepts Chapter

37 . consistent and dependent 4. (5, 0) (5, 0) 5. (4, ) 4 6. (, 4) (4, ) 0 (, 4 7. no solution 8. (0, ) 5 6 (0, ) 9. (0, ) (0, ) 4 0. Consistent and independent; if each equation is written in slopeintercept form, the have different slopes, which means the will intersect at some point.. (, ). (5.5, 0.75) 5. (5, ). (, ) 4., 6. (, ) Glencoe/McGraw-Hill 4 Advanced Mathematical Concepts Chapter

38 7., 8. (5, ) , (5, 6). Sample answer: elimination could be considered easiest since the first equation multiplied b added to the second equation eliminates b; substitution could also be considered easiest since the first equation can be written as a b, making substitution ver eas; (, ). a. 6, 6, 8; 6, 6, 8 b. isosceles 5a. (7, 5.95) 5b. If ou drink 7 servings of soft drink, the price for each option is the same. If ou drink fewer than 7 servings of soft drink during that week, the disposable cup price is better. If ou drink more than 7 servings of soft drink, the refillable mug price is better. See students choices. 5c. ver a ear s time, the refillable mug would be more economical. 7. $ a. B b. Spartans: 4,000; visitors: , a. a d b e 6b. a d, c f b e 6c. a d, c f b e people 40. Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts Chapter

39 A 4. $, {8}, {, }; no, because there are two range values paired with a single domain value. - Solving Sstems of Equations in Three Variables Pages Solving a sstem of three equations involves eliminating one variable to form two sstems of two equations. Then solving is the same.. Sample answer: Use one equation to eliminate one of the variables from the other two equations. Then eliminate one of the remaining variables from the resulting equations. Solve for a variable and substitute to find the values of the other variables. 5. (7,, ) 7. acceleration: ft/s, initial velocit: 56 ft/s, initial height: 5 ft 9. (,, 4). (6, 4, 7). no solution 5. (, 7, 4) 7. (4, 0, 7) 9. International Fund $00; Fied Assets Fund $00; compan stock $600. The solution would be an equation in two variables. Sample eample: the sstem 4 6z, z 6, and 5 6z 7 has a solution of all values of and that satisf (,, ) 6. no solution 8. (, 4, 4) 0. (0, 7, ). (, 7, ) 4. (5,, 4) 6. (0.5, 0.75, 0.) 8. (6, 4, ) 0a. Sample answer: z 5; z ; z 7 Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts Chapter

40 . (, 8, ). (,, ), (,, ) 5. 0b. Sample answer: 4 z ; 4 z 0; 5 z 9 0c. Sample answer: z 6; z 8; z a. Sample answer: a sstem has no solution when ou reach a contradiction, such as 0, as ou solve. b. Sample answer: a sstem has an infinite number of solutions when ou reduce the sstem to two equivalent equations such as and. 4. (45, 60) 6. AB BC CD AD 5 units; ABCD is rhombus. 4 Slope of AB and slope of BC 4, so AB BC. A rhombus with a right angle is a square. Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts Chapter

41 7a. C() b. $000, $50 7c. c() 4 8. C Cost ($000) c() Televisions Produced - Using Matrices to Model Real-World Data Pages Sample answer: Pain Blow Film Reliever Drer Atlanta $4.0 $6.78 $8.98 LA $4. $7.4 $0.49 Meico $.97 $7.4 $.5 Cit Toko $7.08 $6.57 $6.7. The sum of two matrices eists if the matrices have the same dimensions. 5. (7, ) 7. (4, 0) 9. impossible Anthon is correct. A third order matri has rows and columns. This matri has 4 rows and columns. 6. (5, ) impossible Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts Chapter

42 . [6 8] 5. (6, ) 7. (5,.5) 9. (7, 9). (5, 5). (, ) 5. (5,, ) impossible impossible Budget Viewers ($ million) (million) Soft drink Pkg. deliver.9.9 Telecommun (, ) 8. (4, ) 0. (, ). (, ) 4. (, 6) 6. (, 0,, ) 8. impossible Glencoe/McGraw-Hill 9 Advanced Mathematical Concepts Chapter

43 Sample answer: to to 4 0, to to to and older 5a. a, b 0, c 0, d 5b. a matri equal to the original matri 5. The numbers in the first row are the triangular numbers. If ou look at the diagonals in the matri, the triangular numbers are the end numbers. To find the diagonal that contains 00, find the smallest triangular number that is greater than or equal to 00. The formula for the nth n(n ) triangular number is a. TV Radio Record. Classical 0 Jazz 0 pera 0 Musicals b. Classical radio and TV musicals 5a. [4 59 8] 5b. Jul, $479.94; Aug, $409.50; Sep, $ a. A B C D A B 0 C 0 0 D b. No; since the matri shows the number of nodes and the numbers of edges between each pair of nodes, onl equivalent graphs will have the same matri. Solve n(n ) 00. The solution is 6. So the 6rd entr 6(6 ) in the first row is 06. Since , we must count 5 places backward along the diagonal to locate 00 in the matri. This movement takes us from the position (row, column) (, 6) to ( 5, 6 5) (6, 48) ,, 56. consistent and dependent Glencoe/McGraw-Hill 40 Advanced Mathematical Concepts Chapter

44 Sample answer: using (60, 8) and (0, 65), A -4 Modeling Motion with Matrices Pages Translation, reflection, rotation, dilation; translations do not affect the shape, size, or orientation of figures; reflections and rotations do not change the shape or size of figures; dilations do not change the shape, but do change the size of figures.. 90 counterclockwise (60 90) or 70 clockwise; 80 counterclockwise (60 80) or 80 clockwise; 70 counterclockwise (60 70) or 90 clockwise Glencoe/McGraw-Hill 4 Advanced Mathematical Concepts Chapter

45 . Sample answer: the first row of the reflection matri affects the -coordinates and the second row affects the -coordinates. A reflection over the -ais changes (, ) to (, ), so the first row needs to be [ 0] so that the is unchanged, and the second row needs to be [0 ] so that the -coordinates are the opposite. Similar reasoning can be used for a reflection over the -ais, which changes (, ) to (, ) and a reflection over the line, which interchanges the values for and. 5. J (, 7.5), K (.5, 4.5), L (0, ) 4a. 6 4b. 4c. 4d A (, ), B (, ), C (, ), D (, ) J J L L K K A D A D B C B C 7. A (, ), B (4, ), C (, ), D (0, ) B C A D D A C B 8. P (, ), Q (4, ), R (, ) Q P Q R R P Glencoe/McGraw-Hill 4 Advanced Mathematical Concepts Chapter

46 9. L (6, 4), M (, ), N (, ) 0a.. A (, ), B (, ), C (5, ) 0b. 4. X (0, 6), Y (, 6 ), 4 4 Z (, ) 4. P (6, 0), Q (4, 4), R (, 6), S (8, 4) 4a. Glencoe/McGraw-Hill 4 Advanced Mathematical Concepts Chapter

47 Glencoe/McGraw-Hill 44 Advanced Mathematical Concepts Chapter 5. W (, ), X (4, ), Y (6, ) 7. C (0, 5), D (4, 9), E (8, 5), F (4, ) C D E E F F D C W W Y Y X X 4b. 4c. The final results are the same image. 6. (, ), P (, 4), Q (, 6), R (, ) 8a b. 8c. translation of 5 units left and units up G F H G F H G F H Q Q R R P P B C A A D D D C C B B A

48 9. A (, ), B (0, 4), C (, ) 0. D (, 4), E (6, ), F (, 4), G (, ). H (, ), I (, ), J (5, ), K (4, ). L (, ), M (, ), N (, ). (0, 0), P (4, 0), Q (4, 4), R (0, 4) 4. S (, ), T (, ), U (, 5), V (4, 4), W (4, ) Glencoe/McGraw-Hill 45 Advanced Mathematical Concepts Chapter

49 5a. Let R -ais. a b a b a b a b c d c d c d a b c d a b c d a b c d Thus, a, b 0, c 0, and d. B substitution, R -ais c. Let R. a b c d c d a b a b a b c d c d c d a b c d a b c d a b a b a b c d c d c d Thus, a 0, b, c, and d 0. B substitution, R b. Let R -ais. a b a b a b a b c d c d c d a b c d a b c d a b c d Thus, a, b 0, c 0, and d. B substitution, R -ais d. Let Rot 90. a b c d c d a b c d a b c d a b a b a b c d c d c d a b a b a b c d c d c d Thus, a 0, b, c, and d 0. B substitution, Rot Glencoe/McGraw-Hill 46 Advanced Mathematical Concepts Chapter

50 a b 5e. Let c d Rot 80. 5f. Let c d Rot 70. a b a b c d c d a b a b a b a b a b a b c d c d c d c d c d c d a b a b a b c d c d c d Thus, a, b 0, c 0, and d. B substitution, Rot a b a b a b a b c d c d c d Thus, a 0, b, c, and d 0. B substitution, Rot J (4, ), K (, ), L (, 7) 7. J (6, 4), K (, ), L (, ) 8. J (4, 6), K (, ), L (, ) Glencoe/McGraw-Hill 47 Advanced Mathematical Concepts Chapter

51 9a. The bishop moves along a diagonal until it encounters the edge of the board or another piece. The line along which it moves changes verticall and horizontall b unit with each square moved, so the translation matrices are scalars. Sample matrices are c, c, c, and c, where c is the number of squares moved. 9c. The king can move unit in an direction. The matrices describing this are 0 0,, 0 0, 0 0, 0 0,,, and.. (0, 5); (5, 0), (0, 5), (5, 0). See students work; the repeated dilations animate the growth of something from small to larger similar to a lens zooming into the origin. 9b. The knight moves in combinations of vertical- horizontal or vertical- horizontal squares. These can be either up or down, left or right. Sample matrices are,,,,,,, and. 0. Consider a b c d. Dilation with scale factor a b a b c d c d Rotation of 80 0 a b a b 0 c d c d The verte matrices for the images of a dilation with scale factor and a rotation of 80 are the same, so the images are the same.. Sample answer: There is no single matri to achieve this. You could reflect over the -ais and then translate (4) or 8 units upward a. Sample answer: the figure would be enlarged disproportionall. Glencoe/McGraw-Hill 48 Advanced Mathematical Concepts Chapter

52 4b hardbacks $, paperbacks $ , 7 4c. See students work; the figure appears as if blown out of proportion. 6. (0.4,.,.5) 8. (, ) B Chapter Mid-Chapter Quiz Page 96. (4, ) trucks, 000 cars 5.,, , 6 4. (, 5, ) 6. (, 6) 8. impossible 0. The result is the original figure. The original figure is represented a b c 0 a b c d e f 0 d e f b. The reflection over the -ais is found b a b c d e f. The reflection of the image over the -ais is found b 0 a b c a b c. The matri for the final image is the 0 d e f d e f same as that of the original figure. Glencoe/McGraw-Hill 49 Advanced Mathematical Concepts Chapter

53 -5 Determinants and Multiplicative Inverses of Matrices Pages Sample answer: a matri with a nonzero determinant.. Sample answer: ,. 8 kg of the metal with 55% aluminum and kg of the metal with 80% aluminum does not eist Sample answer: is not a square matri; also has no determinant. 4. Sample answer: the sstem has a solution if ad bc 0, since ou can use the inverse of the matri a c b d to find the solution does not eist. (8, 5) does not eist Glencoe/McGraw-Hill 50 Advanced Mathematical Concepts Chapter

54 (, ) 5. (0, ) , 9., 4 4.,, ,4 45. (,, ) a b a b Let A and I. 6. (9, 7) 8. (, ) 4 40.,, (.8, 7.) ,000 in 995 and 40,000 in gal of 0% and 4 gal of 5% A b a b b a b a b a b a a a b a b a b a b AA 0 0 I Thus, AA I. a a b b ab a b ab ba a b a b Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts Chapter

55 a b c d 49. Yes; A or 8.5 square units Does (A ) (A )? A a bc ab bd ac cd bc d (A ) a d abcd b d ad bc A b bc ad c a ad bc ad bc d ad bc (A ) a d abcd b d bc d ac cd bc d ac cd d b c a Thus, (A ) (A ). ab bd a bc ab bd a bc 5. computer sstem: $959, printer: $9 5. H (5, 9), I (, 5), J (, 9), K (, ) 55. infinitel man solutions 5. first test: 86, second test: g() g() ( 5) ( ) or ( ) ( ); 7 Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts Chapter

56 59a. or approimatel , 59b..5 ft 6. No, more than one element of the range is paired with the same element of the domain. 6. B -6 Solving Sstems of Linear Inequalities Pages 09 a. the sum of twice the width and twice the height b. Sample answer: skis, fishing rods. You might epect five vertices; however, if the equations were dependent or if the did not intersect to form the sides of a conve polgon, there would be fewer vertices.. Tomas is correct. There are functions in which the coordinates of more than one verte will ield the same value for the function (, 0), (, ), (0, 4), (7, 0.5), (7, 0) 6. 5, 6 7., 8. at least 00 cards Glencoe/McGraw-Hill 5 Advanced Mathematical Concepts Chapter

57 9. 0. (,0) (0, ) (, ) Yes, it is true for both inequalities () 5 () 6 true 7 true. (0, ), 4,,, 0.5 (0, ) (, ) ( ) , (0, 0), (0, ), (, ) 0 (0, 0) (, ) 0 (0, ) Glencoe/McGraw-Hill 54 Advanced Mathematical Concepts Chapter

58 5. (, 5), (7, 0), (4, 0), (, 5) 6., , 9. 6,. 9, 4. 4, 4, 4, 4 5a. vertices: 5, 0, 6, 0, 8. 4, 0., 9., 5 4. Sample answer:, 4, 4 6. $080 9, 6, 7, 8,, 8,, 4,, 5b. ma at 7, 8 88 ; 5 5 min at, 4 7a. 8a. is $ profit on each batch of garlic dressing and is $ profit on each batch of raspberr dressing. 8b. batches garlic dressing, 4 batches raspberr dressing 7b. f(, ) c. 80 ft at the Main St. site and 0 ft at the High St. site 7d. The maimum number of customers can be reached b renting 0 ft at Main St. Glencoe/McGraw-Hill 55 Advanced Mathematical Concepts Chapter

59 d p. {6}, {4, 4}; no, two -values for one -value. 60 Glencoe/McGraw-Hill 56 Advanced Mathematical Concepts Chapter

60 -7 Linear Programming Pages 5 8. Sample answer: these inequalities are usuall included because in real life, ou cannot make less than 0 of something.. Sample answer: first define variables. Then write the constraints as a sstem of inequalities. Graph the sstem and find the coordinates of the vertices of the polgon formed. Then write an epression to be maimized or minimized. Finall, substitute values from the coordinates of the vertices into the epression and select the greatest or least result. 5a b c.. Sample answer: in an infeasible problem, the region defined b the constraints contains no points. An unbounded region contains an infinite number of points. 4. infeasible brochures, 50 fliers 5d. P(, ) 5 8 5e. 60 small packages, 0 large packages 5f. $800 5g. No; if revenue is maimized, the compan will not deliver an large packages, and customers with large packages to ship will probabl choose another carrier for all of their business. Glencoe/McGraw-Hill 57 Advanced Mathematical Concepts Chapter

61 7. 5 Eplorers, 0 Grande Epeditions 9. infeasible. alternate optimal solutions (, ) 4 (4, ) 8. alternate optimal solutions 0. unbounded a. Let g the number of cups of Good Start food and s the number of cups of Sirius food. 0.84g 0.56s.54 b. 0.g 0.49s 0.56 c. s (, 0)(4, 0) 4 (0,.75) 0.8g 0.56s.54 0.g 0.49s 0.56 (.5, 0.5) (.67, 0) g d. C(g, s) 6g s e. 0 cups of Good Start and.75 cups of Sirius f Glencoe/McGraw-Hill 58 Advanced Mathematical Concepts Chapter

62 a. Let d the number of da-shift workers and n the number of night-shift workers. d 5 n 6 d n 4 b. 4a. 0 acres of corn, 60 acres of sobeans 4b. $,000 c. C(n, d ) 5d 60n d. 8 da-shift and 6 night-shift workers e. $ section-i questions, section- II questions 7. $4000 in First Bank, $7000 in Cit Bank units of snack-size, 800 units of famil-size. alternate optimal solutions a. $70 b. Sample answer: spend more than 0 hours per week on these services. 5. (0, 6) 6. $, nurses, nurse s aides 0. batches of soap and batches of shampoo. 69 square units 4. minimum:, maimum: Sample answer: C $.65 $0.5(n 0); $ A Glencoe/McGraw-Hill 59 Advanced Mathematical Concepts Chapter

63 Chapter The Nature of Graphs - Smmetr and Coordinate Graphs Pages 6. The graph of is an even function. The graph of 6 is an odd function. The graphs of 4 and are neither. a. Sample answer: 0, 0,, ; b. infinitel man c. point smmetr about the origin 5. Alicia; Graphicall: If a graph has origin smmetr, then an portion of the graph in Quadrant I has an image in Quadrant III. If the graph is then smmetric with respect to the -ais, the portion in Quadrants I and II have reflections in Quadrants II and IV, respectivel. Therefore, an piece in Quadrant I has a reflection in Quadrant IV and the same is true for Quadrants II and III. Therefore, the graph is smmetric with respect to the -ais. Algebraicall: Substituting (, ) into the equation followed b substituting (, ) is the same as substituting (, ). 7. es. The graph of an odd function is smmetric with respect to the origin. Therefore, rotating the graph 80 will have no effect on its appearance. See students work for eample. 4. Substitute (a, b) into the equation. Substitute (b, a) into the equation. Check to see if both substitutions result in equivalent equations. 6. no 8. -ais Glencoe/McGraw-Hill 60 Advanced Mathematical Concepts Chapter

64 ais. -ais. -intercept: (5, 0); other points: 6,, 6, 5 5, 5. no 7. es 9. no 6, 5 4. es 6. no 8. es 0. es; g() Replace with. () g() () g() Determine the opposite of the function. g() g() g() Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts Chapter

65 . and. none of these 5. all 7. -ais and -ais, and. -ais 4. -ais 6. -ais and -ais 8. (4, 4) (, ) (, ) (, ) (, ) ( 4, 4) 9. ( 4, 4) (4, 4) 0. Sample answer: (, ) (, ) (4, 4) (, ) (, ) (, ) (, ) (, ). both. -ais. -ais 4. both Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts Chapter

66 5. both 6. neither 7. The equation is smmetric about the -ais. 8a. origin, - and -ais smmetr 8b. 9. Sample answer: 0 8c. (, 5), (, 5), (, 5) 40. Sample answer: Glencoe/McGraw-Hill 6 Advanced Mathematical Concepts Chapter

67 4. (4, 6) or (4, 6) 4. No. If an odd function has a -intercept, then it must be the origin. If it were not, sa it were (0, ), then the graph would have to contain (, 0). This would cause the relationship to fail the vertical line test and it would therefore not be a function. But, not all odd functions have a -intercept. Consider the graph of biccles, 75 triccles 45. (,, 7) consistent and dependent , B Glencoe/McGraw-Hill 64 Advanced Mathematical Concepts Chapter

68 - Families of Graphs Pages ( 4) 7. reflections and translations 5a. g() 5b. h() 5c. k() 7. The graph of g() is the graph of f () compressed horizontall b a factor of, and then reflected over the -ais. 9a. translated up units, portion of graph below -ais reflected over the -ais 9b. reflected over the -ais, compressed horizontall b a factor of 9c. translated left unit, compressed verticall b a factor of The graph of ( ) is a translation of three units to the left. The graph of is a translation of three units up. 4. When c, the graph of f () is compressed horizontall b a factor of c. When c, the graph of f () is unchanged. When 0 c, the graph is epanded horizontall b a factor of. c 6. The graph of g() is the graph of f () translated left 4 units. 8a. epanded horizontall b a factor of 5 8b. translated right 5 units and down units 8c. epanded verticall b a factor of, translated up 6 units 0. Glencoe/McGraw-Hill 65 Advanced Mathematical Concepts Chapter

69 .. The graph of g() is a translation of the graph of f () up 6 units. 5. The graph of g() is the graph of f () compressed horizontall b a factor of The graph of g() is the graph of f () epanded verticall b a factor of. 9. The graph of g() is the graph of f () reflected over the -ais, epanded horizontall b a factor of.5, translated up units. a. b. $50 $00 $50 $00 $ Time (h) $50 $00 $50 $00 $ Time (h) c. $5 4. The graph of g() is the graph of f () compressed verticall b a factor of The graph of g() is a translation of f () right 5 units. 8. The graph of g() is the graph of f () reflected over the -ais. 0a. reflected over the -ais, compressed horizontall b a factor of 0.6 0b. translated right units, epanded verticall b a factor of 4 0c. compressed verticall b a factor of, translated down 5 units Glencoe/McGraw-Hill 66 Advanced Mathematical Concepts Chapter

70 a. epanded horizontall b a factor of 5 b. epanded verticall b a factor of 7, translated down 0.4 units c. reflected across the -ais, translated left unit, epanded verticall b a factor of 9 a. translated left units, compressed verticall b a factor of b. reflected over the -ais, translated down 7 units c. epanded verticall b a factor of, translated right units and up 4 units 5a. compressed horizontall b a factor of, translated down 5 units 5b. reflected over the -ais, compressed verticall b a factor of c. The portion of the parent graph on the left of the -ais is replaced b a reflection of the portion on the right of the -ais. The new image is then translated 4 units right a. translated left units and down 5 units b. epanded horizontall b a factor of.5, reflected over the -ais c. compressed horizontall b a factor of, translated up units 4a. epanded horizontall b a factor of 4b. compressed horizontall b a factor of, translated 8 units up 4c. The portion of the parent graph on the left of the -ais is replaced b a reflection of the portion on the right of the -ais Glencoe/McGraw-Hill 67 Advanced Mathematical Concepts Chapter

71 f( ) f() 8 5a. 0 6a. 0 [7.6, 7.6] scl: b [5, 5] scl: [7.6, 7.6] scl: b [5, 5] scl: Glencoe/McGraw-Hill 68 Advanced Mathematical Concepts Chapter

72 5b b c..5 [7.6, 7.6] scl: b [5, 5] scl: [7.6, 7.6] scl: b [5, 5] scl: 6c a. 0 [7.6, 7.6] scl: b [5, 5] scl: [7.6, 7.6] scl: b [5, 5] scl: 8a. The graph would continuall move left units and down units. 8b. The graph would continuall be reflected over the -ais and moved right unit. 7b..5 [7.6, 7.6] scl: b [5, 5] scl: [7.6, 7.6] scl: b [5, 5] scl: Glencoe/McGraw-Hill 69 Advanced Mathematical Concepts Chapter

73 7c. 0.6 [7.6, 7.6] scl: b [5, 5] scl: b 9. The -intercept will be. 40a. a 0.5[[ ]].50 if [[]] 0.5[[]].50 if [[]] 40b. 4a. 5 units Price (dollars) 0 0 4a. () () () (4) 4 5 Fare Units 4b The area of the triangle is (0)(0) or 50 units. Its area 4b. () () is twice as large as that of the original triangle. The area of the triangle formed b c f() would be 5c units. Glencoe/McGraw-Hill 70 Advanced Mathematical Concepts Chapter

74 4c. 4b. (cont d.) () (4) The area of the triangle is (0)(5) or 5 units. Its area is the same as that of the original triangle. The area of the triangle formed b f( c) would be 5 units. 4a. reflection over the -ais, reflection over the -ais, vertical translation, horizontal compression or epansion, and vertical epansion or compression 4b. horizontal translation preschoolers and 0 school-age 47. ±5, 9, z The graph implies a negative linear relationship A 4c. () ( ) 5 () ( ) 5 () ( ) 5 (4) ( ) es; f() f() 46. A (4, 5), B (, ), C (, ) 48. (, 4) , Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts Chapter

75 - Graphs of Nonlinear Inequalities Pages A knowledge of transformations can help determine the graph of the boundar of the shaded region, 5.. Sample Answer: Pick a point, not on the boundar of the inequalit, and test to see if it is a solution to the inequalit. If that point is a solution, shade all points in that region. If it is not a solution to the inequalit, test a point on the other side of the boundar and shade accordingl.. When solving a one variable inequalit algebraicall, ou must consider the case where the quantit inside the absolute value is non-negative and the case where the quantit inside the absolute value is negative. 4. This inequalit has no solution since the two graphs do not intersect. 5. es no { 0 or } Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts Chapter

76 . { }. no 5. es 7. es 9. (0, 0), (, ), and (, ); if these points are in the shaded region and the other points are not, then the graph is correct. a b..005 cm,.995 cm 4. no 6. no 8. es Glencoe/McGraw-Hill 7 Advanced Mathematical Concepts Chapter

77 { 9 or } 5. { 9} 7. no solution 9. { 7 7} 4. { 5.5 0} { 8 or 0} 6. {.5} 8. all real numbers , units 44a.b 0 44b. none 44c. b 0 or b 4 44d. b 4 44e. 0 b 4 Glencoe/McGraw-Hill 74 Advanced Mathematical Concepts Chapter

78 45a. 46. The graph of g() is the graph of f() reflected over the -ais and epanded verticall b a factor of. 45b. The shaded region shows all points (, ) where represents the number of cookies sold and represents the possible profit made for a given week ais , Glencoe/McGraw-Hill 75 Advanced Mathematical Concepts Chapter

79 Chapter Mid-Chapter Quiz Page 5. -ais, -ais,,, origin.,, origin 5a. translated down units 5b. reflected over the -ais, translated right units 5c. compressed verticall b a factor of, translated up unit none of these 4. -ais 6a. epanded verticall b a factor of 6b. epanded horizontall b a factor of and translated down unit 6c. translated left unit and up 4 units ; Inverse Functions and Relations Pages Sample answer: First, let f (). Then interchange and. Finall, solve the resulting equation for.. Sample answer: f (). n is odd. 4. Sample answer: If ou draw a horizontal line through the graph of the function and it intersects the graph more than once, then the inverse is not a function. Glencoe/McGraw-Hill 76 Advanced Mathematical Concepts Chapter

80 5. She is wrong. The inverse is f () ( ), which is a function f () ; f () is a function.. f () ; 6 f () is not a function. 0. f () ; f () is a function... f () 0; [f f ]() f( 0) ( 0) 5 4a. r 4b..% B 0 [f f]() f Since [f f ]() [f f ](), f and f are inverse functions. Glencoe/McGraw-Hill 77 Advanced Mathematical Concepts Chapter

81 5. f() f() f () 6. f() f() f () 7. f() f() 8. f() f () 0 f() f () f() f () 0. f() f() f() f (). f() 8 f() 4 f (). f() f () f() 8. f() f() f () 4. f () 4 f() f() f () Glencoe/McGraw-Hill 78 Advanced Mathematical Concepts Chapter

82 5. f 7 () ; f () is a function. 7. f () ; f () is a function. 9. f () ; 7 f () is not a function.. f () ; f () is a function.. f () ; f () is a function f () ; f () is a function. 8. f () ; f () is not a function. 0. f () ; f () is not a function.. f () ; f () is not a function Glencoe/McGraw-Hill 79 Advanced Mathematical Concepts Chapter

83 9. f () 40. f () 4 4 [f f ]() f or 6 6 [f f]() f or 4 4 Since [f f ]() [f f ](), f and f are inverse functions. 4a. d () [f f ]() f or [f f]() f ( ) 4 ) [( 4] 4 or Since [f f ]() [f f ](), f and f are inverse functions. v 4a. h 64 4b. Yes. The pump can propel water to a height of about 88 ft. 4b. No; the graph of d() fails the horizontal line test. 4c. d () gives the numbers that are 4 units from on the number line. There are alwas two such numbers, so d associates two values with each -value. Hence, d () is not a function. Glencoe/McGraw-Hill 80 Advanced Mathematical Concepts Chapter

84 4a. Sample answer:. 4b. The graph of the function must be smmetric about the line. 4c. Yes, because the line is the ais of smmetr and the reflection line. 44a. 44b. positive real numbers; positive multiples of 0 44c. 45. It must be translated up 6 units and 5 units to the left; ( 6) 5, a. Yes. If the encoded message is not unique, it ma not decode properl. 47b. The inverse of the encoding function must be a function so that the encoded message ma be decoded. 47c. c () ( ) 47d. FUNCTINS ARE FUN 49. both 44d. positive multiples of 0; positive real numbers. 44e. C () gives the possible lengths of phone calls that cost. KE 46a. v m 46b. 5.5 m/sec 46c. There are alwas two velocities. 48. { 5 } 50a. a 0, b 0, 4a b, a 6b 54 50b. 4 gallons Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts Chapter

85 5. (, 7) neither C -5 Continuit and End Behavior Pages Sample answer: The function approaches as approaches from the left, but the function approaches 4 as approaches from the right. This means the function fails the second condition in the continuit test.. Infinite discontinuit; f () as, as f () as. 5. No; is undefined when.. a n n p() positive even positive even positive odd positive odd negative even negative even negative odd negative odd 4. f () is decreasing for 0 and increasing for 0. g() is increasing for 0 and decreasing for 0. Reflecting a graph switches the monotonicit. In other words, if f () is increasing, the reflection will be decreasing and vice versa. 6. No; f () approaches 6 as approaches from the left but f () approaches 6 as approaches from the right. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts Chapter

86 7. as, as as, as. 0. decreasing for ; increasing for a. t 4 b. when t 4 c. 0 amps. No. The function is undefined when. 5. Yes. The function is defined when ; the function approaches (in fact is equal to ) as approaches from both sides; and when. 7. Yes. The function is defined when ; f () approaches as approaches from both sides; and f (). 9. Sample answer: 0. g() is undefined when 0.. as, as.. as, as. decreasing for and ; increasing for. Yes. The function is defined when ; the function approaches as approaches from both sides; and when. 4. Yes. The function is defined when ; the function approaches 0 as approaches from both sides; and f () No. f () approaches 7 as approaches 4 from the left, but f () approaches 6 as approaches 4 from the right. 8. jump discontinuit 0. as, as.. as, as as, 0 as. Glencoe/McGraw-Hill 8 Advanced Mathematical Concepts Chapter