5144 Demana TE Ans pp895-914 Answers Ch... · 51. (d) 52. (b) 53. (a)(b)(c) The year 2006 is represented by x 16. So the value of y for x 16 is 6527 million, a little larger than

ADDITIONAL ANSWERS 895

Exercises P.15. ; All real numbers less than or equal to 2

6. ; All real numbers greater than or equal to �2 and less than 5

7. ; All real numbers less than 7

8. ; All real numbers between �3 and 3, including �3 and 3

9. ; All real numbers less than 0

10. ; All real numbers between 2 and 6, including 2 and 6

23. The real numbers greater than 4 and less than or equal to 924. The real numbers greater than or equal to �1, or the real numbers which are at least �125. The real numbers greater than or equal to �3, or the real numbers which are at least –326. The real numbers between �5 and 7, or the real numbers greater than �5 and less than 727. The real numbers greater than �128. The real numbers between �3 and 0 (inclusive), or the real numbers greater than or equal to �3 and less than or equal to 0.33. x � 29 or [29, �); x � Bill’s age 34. 0 � x � 2 or [0, 2]; x � cost of an item35. 1.099 � x � 1.399 or [1.099, 1.399]; x � dollars per gallon of gasoline36. 0.02 � x � 0.065 or (0.02, 0.065); x � average percent of all salary raises45. (a) Associative property of multiplication

(b) Commutative property of multiplication(c) Addition inverse property(d) Addition identity property(e) Distributive property of multiplication over addition

46. (a) Multiplication inverse property(b) Multiplication identity property, or distributive property of multiplication over addition, followed by the multiplication identity property. Note that we also use the multiplicative commutative property to say that 1 � u � u � 1 � u.(c) Distributive property of multiplication over subtraction(d) Definition of subtraction; associative property of addition;

definition of subtraction(e) Associative property of multiplication; multiplication

inverse; multiplication identity

0 1 2 3 4 5 6 7 8 9�1

0 1 2 3 4 5�1�2�3�4�5

0 1 2 3 4 5�1�2�3�4�5

0 1 2 3 4 5 6 7 8�1�2

0 1 2 3 4 5�1�2�3�4 6

0 1 2 3 4 5�1�2�3�4�5

Additional Answers

Step Quotient Remainder

1 0 1

2 0 10

3 5 15

4 8 14

5 8 4

6 2 6

7 3 9

8 5 5

9 2 16

10 9 7

11 4 2

12 1 3

13 1 13

14 7 11

15 6 8

16 4 12

17 7 1

66. (a)

5144_Demana_TE_Ans_pp895-914 1/24/06 8:27 AM Page 895

SECTION P.2Quick Review P.21. 2. 3.

Distance: �7� � �2� � 1.232 Distance: �125� or 0.13� 4.

5. 6.

Exercises P.23. (a) First quadrant (b) On the y-axis, between quadrants I and II (c) Second quadrant (d) Third quadrant4. (a) First quadrant (b) On the x-axis, between quadrants II and III (c) Third quadrant (d) Third quadrant29. 30. 31.

32. 33. 34.

37. The three sides have lengths 5, 5, and 5�2�. Since two sides have the same length, the triangle is isosceles.

SECTION P.3Exercises P.329. (a) The figure shows that x � �2 is a solution of the equation 2x2 x � 6 � 0.

(b) The figure shows that x � �32

� is a solution of the equation 2x2 x � 6 � 0.

30. (a) The figure shows that x � 2 is not a solution of the equation 7x 5 � 4x � 7.(b) The figure shows that x � �4 is a solution of the equation 7x 5 � 4x � 7.35. 36. 37.

x � 6 x 2 x � �2

0 1 2 3 4 5�1�2�3�4�50 1 2 3 4 5 6 7 8 9�10 1 2 3 4 5 6 7 8 9�1

[1995, 2005] by [0, 150][1995, 2005] by [0, 50][1995, 2005] by [0, 100]

[1995, 2005] by [0, 150][1995, 2005] by [0, 5][1995, 2005] by [0, 10]

y

x

5

5

AB

C

D

y

x

5

5

A

B

C

D

0 1 2 3 4 5�1�2�3�4�5

0 1 2 3 4 5�1�2�3�4�5�1.75 �1.5�21.50.5 2 2.5 31

896 ADDITIONAL ANSWERS



38. 39. 40.

x � �3 � 4 � x � 3 �13

� � x � 3

41. 42.

x � 3 x � 5

59. Multiply both sides of the first equation by 2. 60. Divide both sides of the first equation by 2.63. False. �6 � �2 because �6 lies to the left of �2 on the number line.

64. True. 2 � �63

� includes the possibility that 2 � �63

�, which is true. 69. (e) If your calculator returns 0 when you enter

2x 1 � 4, you can conclude that the value stored in x is not a solution of the inequality 2x 1 � 4.

SECTION P.4Exploration 11. The graphs of y � mx b and y � mx c 2.

have the same slope but different y-intercepts.

The angle between the two lines appears to be 90°.

3.

In each case, the two lines appear to be at right angles to one another.

Exercises P.427. 28. 29. 30.

41. (a) y � 3x � 1 (b) y � ��13

�x �73

�

42. (a) y � �2x � 1 (b) y � �12

�x 4 43. (a) y � ��23

�x 3 (b) y � �32

�x � �72

� 44. (a) y � �35

�x � �153� (b) y � ��

53

�x 11

49. m � �38

� �142�, so asphalt shingles are acceptable. 50. Americans’ income was, respectively, 2000: 8.2, 2002: 9, 2003: 9.4 trillion dollars.

[–1, 3] by [–50, 350][–1, 5] by [–10, 80][–5, 20] by [–10, 40][–5, 10] by [–10, 60]

[–4.7, 4.7] by [–3.1, 3.1]m=5

[–4.7, 4.7] by [–3.1, 3.1]m=4

[–4.7, 4.7] by [–3.1, 3.1]m=3

[–4.7, 4.7] by [–3.1, 3.1]m=1

[–4.7, 4.7] by [–3.1, 3.1]

0 1 2 3 4 5 6 7 8 9�10 1 2 3 4 5 6 7 8�1�2

0 1 2 3 4 5�1�2�3�4�50 1 2 3 4 5�1�2�3�4�50 1 2 3 4 5�1�2�3�4�5


51. (d) 52. (b)

53. (a) (b) (c) The year 2006 is represented by x � 16. So the value of y for x � 16is � 6527 million, a little larger than the U.S. Census Bureau esti-mate of 6525 million.

54. (a) (b)

59. (a) No; perpendicular lines have slopes with opposite signs. (b) No; perpendicular lines have slopes with opposite signs.

60. (a) If b � 0, both lines are vertical; otherwise, both have slope m � ��ab

�. If c � d, the lines are coincident.

(b) If either a or b equals 0, then one line is horizontal and the other is vertical. Otherwise, their slopes are ��ab

� and �ba

�, respectively.

In either case, they are perpendicular. 61. False. The slope of a vertical line is undefined. For example, the vertical line through (3, 1) and (3, 6) would have slope (6 � 1)/(3 � 3) � 5/0, which is undefined. 62. True. If b � 0, then a � 0 and the graph of

x � �ac

� is a vertical line. If b � 0, then y � ��ab

�x �bc

� is a line with slope ��ab

� and y-intercept �bc

�.

67. (a) (b) (c) (d) a is the x-intercept and b isthe y-intercept when c � 1.

(e) (f) When c � �1, a is the opposite of the x-intercept and b is the opposite of the y-intercept.

a is half the x-intercept and b is half the y-intercept when c � 2.

[–10, 10] by [–10, 10][–10, 10] by [–10, 10][–10, 10] by [–10, 10]

[–5, 5] by [–5, 5][–5, 5] by [–5, 5][–5, 5] by [–5, 5]

[0, 15] by [0, 100][0, 15] by [0, 100]

[0, 15] by [5000, 7000][0, 15] by [5000, 7000]

[1995, 2005] by [50, 180][1995, 2005] by [5, 10]




68. (a) (b) If m 0, then the graphs of y � mx and y � �mx have the same steepness, but one increases from left to right, and the other decreases from left to right.

(c) These graphs have the same slope, but different y-intercepts.

69. As in the diagram, we can choose one point to be the origin, and another to be on the x-axis. The midpoints of the sides, starting

from the origin and working around counterclockwise in the diagram, are then A��a2

�, 0�, B��a

2b

�, �2c

��, C��b

2d

�, �c

2e

��, and D��d2

�, �2e

��. The opposite sides are therefore parallel, since the slopes of the four lines connecting those points are: mAB � �

bc

�; mBC � �d �

ea

�;

mCD � �bc

�; mDA � �d �

ea

�. 70. y � 4 � ��34

�(x � 3) 71. A has coordinates ��b2

�, �2c

��, while B is ��a

2b

�, �2c

��, so the line

containing A and B is the horizontal line y � �2c

�, and the distance from A to B is ��a

2b

� � �b2

�� a2

�.

SECTION P.5Exploration 11. 3.

By this method, we have zeros at 0.79 and 2.21.

4. 5. The answers in parts 2, 3, and 4 are the same.6. On a calculator, evaluating 4x2 � 12x 7 when x � 0.79 gives

y � 0.0164 and when x � 2.21 gives y � 0.0164, so the numbers 0.79and 2.21 are approximate zeros.

Zooming in and tracing reveals the same zeros, correct to 2 decimal places.

Exercises P.51. x � �4 or x � 5 3. x � 0.5 or x � 1.5 5. x � ��

23

� or x � 3 9. x � �4 � ��83

�� 11. y � ��72

��13. x � �7 or x � 1 14. x � �2.5 � �1�5�.2�5� � �6.41 or x � �2.5 �1�5�.2�5� � 1.41

15. x � �72

� � �1�1� � 0.18 or x � �72

� �1�1� � 6.82 16. x � �3 � �1�3� � �6.61 or x � �3 �1�3� � 0.61

18. x � �43

� � �13

��4�6� � �0.93 or x � �43

� �13

��4�6� � 3.59 19. x � �4 � 3�2� � �8.24 or x � �4 3�2� � 0.24

[0.63, 0.94] by [–0.39, 0.55][2.05, 2.36] by [–0.5, 0.43]

[–1, 4] by [–5, 10][–1, 4] by [–5, 10][–1, 4] by [–5, 10]

[–8, 8] by [–5, 5]

[–8, 8] by [–5, 5]

These graphs all pass through the origin. They have different slopes.


20. x � �12

� or x � 1 21. x � �1 or x � 4 22. x � �12

��3� � �12

��2�3� � �1.53 or x � �12

��3� �12

��23� � 3.26

23. ��52

� ��

273�� 1.77 or ��

52

� � ��

273�� 6.77

29. 30.

31. 32.

33. 34. 35. x2 2x � 1 � 0; x � 0.4 36. x3 � 3x � 0; x � �1.73

45. (a) y1 � 3�x�� 4� (the one that begins on the x-axis) and y2 � x2 � 1 (b) y � 3�x�� 4� � x2 1(c) The x-coordinates of the intersections in the first picture are the same as the x-coordinates where the second graph crosses the x-axis.46. Any number between 1.324 and 1.325 must have the digit 4 in its thousandths position. Such a number would round to 1.32.47. x � �2 or x � 1 48. x � �4.24 or x � 4.24 49. x � 3 or x � �2 50. x � �8� � 2.8357. (a) There must be 2 distinct real zeros, because b2 � 4ac 0 implies that ��b�2�� 4�a�c� are 2 distinct real numbers.

(b) There must be 1 real zero, because b2 � 4ac � 0 implies that ��b�2�� 4�a�c� � 0, so the root must be x � ��2ba�.

(c) There must be no real zeros, because b2 � 4ac � 0 implies that ��b�2�� 4�a�c� are not real numbers.62. True. If 2 is an x-intercept, then y � 0 when x � 2. That is, ax2 bx c � 0 when x � 2.63. False. Notice that 2(�3)2 � 18, so x could also be �3.

SECTION P.6 Exercises P.652. The graph either lies entirely above the x-axis or entirely below the x-axis.

53. (a bi) � (a � bi) � 2bi, real part is zero

54. (a bi) � (�a��b�i�)� � (a bi) � (a � bi) � a2 b2, imaginary part is zero

55. (�a��b�i�)��(�c��d�i�)� � (�a�c��b�d�)��(�a�d��b�c�)�i� � (ac � bd) � (ad bc)i and

(�a��b�i�)��(�c��d�i�)� � (a � bi) � (c � di) � (ac � bd) � (ad bc)i are equal

56. (�a��b�i�)��(�c��d�i�)� � (�a��c�)��(�b��d�)�i� � (a c) � (b d)i and

(�a��b�i�)� (�c��d�i�)� � (a � bi) (c � di) � (a c) � (b d)i are equal

[–5, 5] by [–5, 5][–5, 5] by [–5, 5]

[–5, 5] by [–5, 5][–5, 5] by [–5, 5][–5, 5] by [–5, 5][–5, 5] by [–5, 5]

[–5, 5] by [–5, 5][–5, 5] by [–5, 5][–5, 5] by [–5, 5][–5, 5] by [–5, 5]




SECTION P.7Exercises P.71. 2. 3.

4. 5. 6.

7. 8.

11. (��, �5) � ��32

�, �� 13. (��, �2) � ��13

�, ��18. ��, �

34

� � �43

�, �� 19. ��, ��12

�� 43

�, ��21. (��, �1.41] � [0.08, �) 23. ��, �

12

�� 12

�, ��35. Reveals the boundaries of the solution set 37. (b) When x is in the interval (1, 25].

CHAPTER P REVIEW EXERCISES1. Endpoints 0 and 5; bounded 10. 0.000 000 000 000 000 000 000 000 000 910 94

(27 zeros between the decimal point and the 9) 14. (a) � 9.85 (b) ��12

�, 1� 17. (x � 0)2 (y � 0)2 � 22, or x2 y2 � 4

18. (x � 5)2 [y � (�3)]2 � 42, or (x � 5)2 (y 3)2 � 16 19. Center: (�5, �4); radius: 3 28. y � �32

�x � �52

�

33. (a) (b) y � 1.6x 498 (c) 507.6, which is very close to 508.

36. Both graphs look the same, but the graph on the left has slope �23

� — less than the slope of the one on the right, which is �1125� � �

45

�.

The different horizontal and vertical scales for the two windows make it difficult to judge by looking at the graphs.

59. (�6, 3] 60. ��53

�, ��

SECTION 1.1 Exploration 23. A statistician might look for adverse economic factors in 1990, especially those that would affect people near or below the poverty line.4. Yes. Table 1.1 shows that the minimum wage worker had less purchasing power in 1990 than in any other year since 1950.

0 1 2 3 4 5�1�2�3�4�50 2 4 6 8 10�2�4�6�8�10

[0, 15] by [500, 525]

[0, 15] by [500, 525]

0 10 20 30 40 50�10�20�30�40�50�8 �6 �4 �2 0 2 4 6 8�10�12

0 1 2 3 4 5�1�2�3�4�50 1 2 3 4 5�1�2�3�4�5�8 �6 �4 �2 0 2 4 6 8�10�12

0 1 2 3 4 5 6 7 8�1�20 1 2 3 4 5�1�2�3�4�5�8 �6 �4 �2 0 2 4 6 8�10�12


Exercises 1.113. Women (�), Men () 19. (a) and (b)

22. (a) � 3.35 sec24. (a) (b) The line is y � 23.76x 452.3. (c) About 2010.

(d) The terrorist attacks on September 11,2001 caused a major disruption inAmerican air traffic, from which theairline industry was slow to recover.

39. 40. 41. 42.

x � 3.91x � �1.09 or x � 2.86

x � 1.33 or x � 4x � 2.66

43. 44. 45. 46.

x � 1.77x � 2.36

x � �1.47

48. (c) A vertical line through the x-intercept of y3 passes through the point of intersection of y1 and y2.(d) At x � 1.6813306, y1 � y2 � 11.725322. At x � �0.3579264, y1 � y2 � 3.5682944. At x � �3.323404, y1 � y2 � �8.293616.

49. (b)

53. Let n be any integer. n2 2n � n(n 2), which is either the product of two odd integers or the product of two even integers. The product of two odd integers is odd. The product of two even integers is a multiple of 4, since each even integer in the product con-tributes a factor of 2 to the product. Therefore, n2 2n is either odd or a multiple of 4.

[0, 1] by [0, 1]

[–3, 3] by [–1, 4][–4, 4] by [–10, 10][–5, 5] by [–10, 10]

[–5, 5] by [–10, 10]

[–10, 10] by [–2, 2][–10, 10] by [–10, 10]

[–10, 10] by [–10, 10][–10, 10] by [–10, 10]

[–1, 15] by [400, 750][–1, 15] by [400, 750]

[–5, 55] by [23, 92]


L3

3.79754.3755.54055.89866.657


62. (a)

63. (a) (c) The fit is very good:

(e)

SECTION 1.2 Exploration 11. From left to right, the tables are (c) constant, (b) decreasing, and (a) increasing.

2. 3. positive; negative; 04. For lines, Y/ X is the slope.

Lines with positive slope are increasing, lines with negative slope are decreasing, and lines with 0 slope are constant.

Exercises 1.22. Not a function; y has two values for each value of x. 3. Not a function; y has two values for each positive value of x.9. 10. 11. 12.

[–10, 10] by [–10, 10][–10, 10] by [–10, 10][–5, 15] by [–10, 10][–5, 5] by [–5, 15]

[4, 15] by [10, 200] [4, 15] by [30, 60]

Subscribers Monthly Bills

[7, 15] by [50, 200][7, 15] by [50, 200] [7, 15] by [35, 55]

Subscribers Monthly Bills

[–4, 4] by [–10, 10]


X movesfrom

X Y1

�2 to �1 1 0

�1 to 0 1 0

0 to 1 1 0

1 to 3 2 0

3 to 7 4 0

X movesfrom

X Y2

�2 to �1 1 �2

�1 to 0 1 �1

0 to 1 1 �2

1 to 3 2 �4

3 to 7 4 �6

X movesfrom

X Y3

�2 to �1 1 2

�1 to 0 1 2

0 to 1 1 2

1 to 3 2 3

3 to 7 4 6


13. 14. 15. 16.

21. Yes, non-removable 22. Yes, removable 23. Yes, non-removable 24. Yes, non-removable

25. Local maxima at (�1, 4) and (5, 5), local minimum at (2, 2). The function increases on (��, �1], decreases on [�1, 2],increases on [2, 5], and decreases on [5, �). 26. Local minimum at (1, 2), (3, 3) is neither, and (5, 7) is a local maximum. The function decreases on (��, 1], increases on [1, 5], and decreases on [5, �). 27. (�1, 3) and (3, 3) are neither, (1, 5) is a local maximum, and (5, 1) is a local minimum. The function increases on (��, 1], decreases on [1, 5], and increases on [5, �).28. (�1, 1) and (3, 1) are local minima, while (1, 6) and (5, 4) are local maxima. The function decreases on (��, �1], increases on [�1, 1],decreases on (1, 3], increases on [3, 5], and decreases on [5, �).

29. Decreasing on (��, �2]; 30. Decreasing on (��, �1]; 31. Decreasing on (��, �2]; 32. Decreasing on (��, �2]; increasing on [�2, �) constant on [�1, 1]; constant on [�2, 1]; increasing on [�2, �)

increasing on [1, �) increasing on [1, �)

33. Increasing on (��, 1]; 34. Increasing on (��, ��0.549]; decreasing on [1, �) decreasing on [��0.549, �1.215];

increasing on [�1.215, �).

[–2, 3] by [–3, 1][–4, 6] by [–25, 25]

[–10, 10] by [0, 20][–10, 10] by [–2, 18][–7, 3] by [–2, 13][–10, 10] by [–2, 18]

[–5, 5] by [–5, 5][–10, 10] by [–2, 2][–5, 5] by [–10, 10][–10, 10] by [–10, 10]

[–10, 10] by [–5, 5] [–5, 5] by [0, 16][–5, 5] by [–5, 5][–3, 3] by [–2, 2]



41. f has a local minimum of y � 3.75 at x � 0.5. 42. Local maximum: y � 4.08 at x � �1.15. It has no maximum. Local minimum: y � �2.08 at x � 1.15.

43. Local minimum: y � �4.09 at x � �0.82. 44. Local minimum: y � 9.48 at x � �1.67. Local maximum: y � �1.91 at x � 0.82. Local maximum: y � 0 when x � 1.

45. Local maximum: y � 9.16 at x � �3.20. 46. Local maximum: y � 0 at x � �2.5. Local minimum: y � 0 at x � 0 and y � 0 at x � �4. Local minimum: y � �3.13 at x � �1.25.

55. 56. 57. 58.

59. 60. 61. 62.

[–6, 4] by [–10, 10][–4, 6] by [–5, 5][–5, 5] by [0, 5][–10, 10] by [–10, 10]

[–10, 10] by [–10, 10][–8, 12] by [–10, 10][–10, 10] by [–10, 10][–10, 10] by [–10, 10]

[–5, 5] by [–10, 10][–5, 5] by [0, 80]

[–5, 5] by [–50, 50][–5, 5] by [–50, 50]

[–5, 5] by [–50, 50][–5, 5] by [0, 36]



67. (a) f(x) crosses the horizontal (b) g(x) crosses the horizontal (c) h(x) intersects the horizontal asymptote at (0, 0). asymptote at (0, 0). asymptote at (0, 0).

69. (a) The vertical asymptote is x � 0, and 70. The horizontal asymptotes are determined by the two limits this function is undefined at x � 0 lim

x → ��f(x) and lim

x → �f(x). These are at most two different

(because a denominator can’t be zero). numbers.(b) 71. True; this is the definition of the graph of a function.

72. False; f(x) � 0 is a function that is symmetric with respect to the x-axis.

Add the point (0, 0).

77. (a) (b) �1

xx2� � 1 ⇔ x � 1 x2 ⇔ x2 � x 1 0; but the discriminant of x2 � x 1

is negative (�3), so the graph never crosses the x-axis on the interval (0, �).

(d) �1

xx2� �1 ⇔ x �1 � x2 ⇔ x2 x 1 0; but the discriminant of x2 � x 1

is negative (�3), so the graph never crosses the x-axis on the interval (��, 0).

78. (a) increasing (b) (c) ; y is none of these since it first increases from 1 to 1.05 and then decreases.

86.

[–6, 6] by [–2, 2]

[–3, 3] by [–2, 2]

[–10, 10] by [–10, 10]

[–5, 5] by [–5, 5][–10, 10] by [–5, 5][–10, 10] by [–10, 10]


y

0.05

�0.53

�0.09

�0.07

�0.03

�0.02

�0.03

y

1

1.05

0.52

0.43

0.36

0.33

0.31

0.28


SECTION 1.3 Exercises 1.329. Domain: all reals; Range: [�5, �) 30. Domain: all reals; Range: [0, �) 31. Domain: (�6, �); Range: all reals32. Domain: (��, 0) � (0, �); Range: (��, 3) � (3, �) 33. Domain: all reals; Range: all integers34. Domain: all reals; Range: [0, �) 35. (a) Increasing on [10, �) (b) Neither (c) Minimum value of 0 at x � 10(d) Square root function, shifted 10 units right 36. (a) Increasing on [(2k � 1)�/2, (2k 1)�/2] and decreasing on [(2k 1)�/2,(2k 3)�/2], where k is an even integer (b) Neither (c) Minimum of 4 at (2k � 1)�/2 and maximum of 6 at (2k 1)�/2, where k is an eveninteger (d) Sine function shifted 5 units up 37. (a) Increasing on (��, �) (b) Neither (c) None (d) Logistic function, stretched verti-cally by a factor of 3 38. (a) Increasing on (��, �) (b) Neither (c) None (d) Exponential function, shifted 2 units up 39. (a)Increasing on [0, �); decreasing on (��, 0] (b) Even (c) Minimum of �10 at x � 0 (d) Absolute value function, shifted 10 units down40. (a) Increasing on [(2k � 1)�, 2k�] and decreasing on [2k�, (2k 1)�], where k is an integer (b) Even (c) Minimum of �4 at (2k � 1)�and maximum of 4 at 2k�, where k is an integer (d) Cosine function, stretched vertically by a factor of 4 41. (a) Increasing on [2, �);decreasing on (��, 2] (b) Neither (c) Minimum of 0 at x � 2 (d) Absolute value function, shifted 2 units right 42. (a) Increasing on(��, 0]; decreasing on [0, �) (b) Even (c) Maximum of 5 at x � 0 (d) Absolute value function, reflected across x-axis and then shifted5 units up45. 46. 47. 48.

No points of discontinuity x � 0 No points of discontinuity x � 0

49. 50. 51. 52.

No points of discontinuity No points of discontinuity x � 0 x � 2, 3, 4, 5, ...

53. (a) 55. (a)

f(x) � x

(b) The fact that ln(ex) � x shows that the natural logarithmfunction takes on arbitrarily large values. In particular, it takeson the value L when x � eL.

[–5, 5] by [–5, 5][–5, 5] by [–5, 5]

x

y

5

5

x

y

5

5

x

y

5

5

x

y

5

5

x

y

5

5

x

y

5

5

x

y

5

5

x

y

5

5



56. (a)

(b) One possible answer: It is similar because it has discontinuities spaced at regular intervals. It is different because its domain is the set of positive real numbers, and because it is constant on intervals of the form (k, k 1] instead of [k, k 1), where k is an integer.

57. Domain: all real numbers; Range: all integers; Continuity: There is a discontinuity at each integer value of x; Increasing/decreasing behavior: constant on intervals of the form [k, k 1), where k is an integer; Symmetry: none; Boundedness: not bounded; Local extrema: every non-integer is both a local minimum and local maximum; Horizontal asymptotres: none; Vertical asymptotes: none; End behavior: int(x) → �� as x → �� and int(x) → � as x → �.64. (a) Answers will vary.

(b) In this window, it appears that �x� � x � x2. (c) (d) On the interval (0, 1), x2 � x � �x�. On the interval (1, �), �x� � x � x2. All three functions equal 1 when x � 1.

66. Answers will vary. 67. (a) Pepperoni count ought to be proportional to the area of the pizza, which is proportional to the square of the radius.

SECTION 1.4 Exploration 1

[0, 2] by [0, 1.5]

x

y

543

Weight (oz)

21

1.29

1.06

0.83

Cos

t ($)

0.60

0.37


f g f � g

2x � 3 �x

23

� x

�2x 4� �(x � 2)

2(x 2)� x2

�x� x2 �x�

x5 x0.6 x3

x � 3 ln(e3x) ln x

2 sin x cos x �2x

� sin x

1 � 2x2 sin��2x

�� cos x


Exercises 1.41. ( f g)(x) � 2x � 1 x2; ( f � g)(x) � 2x � 1 � x2; ( fg)(x) � (2x � 1)(x2) � 2x3 � x2. There are no restrictions on any of the domains, soall three domains are (��, �).2. ( f g)(x) � x2 � 3x 4; ( f � g)(x) � x2 � x � 2; ( fg)(x) � (x � 1)2(3 � x) � �x3 5x2 � 7x 3. There are no restrictions on any of thedomains, so all three domains are (��, �).3. ( f g)(x) � �x� sin x; ( f � g)(x) � �x� � sin x; ( fg)(x) � �x� sin x. Domain in each case is [0, �).4. ( f g)(x) � �x�� 5� �x 3�; ( f � g)(x) � �x�� 5� � �x 3�; ( fg)(x) � �x�� 5� �x 3�. Domain in each case is [�5, �).

5. ( f/g)(x) � ; x 3 � 0 and x � 0, so the domain is [�3, 0) � (0, �).

(g/f )(x) � ; x 3 0, so the domain is (�3, �).

6. ( f/g)(x) � � ��xx

�

24

��; x � 2 � 0 and x 4 0, so x � 2 and x �4; the domain is [2, �).

(g/f )(x) � � ��xx

�

42

��; x � 2 0 and x 4 � 0, so x 2 and x � �4; the domain is (2, �).

7. ( f /g)(x) � x2 / �1 � x2�; 1 � x2 0, so x2 � 1; the domain is (�1, 1).

( g/f )(x) � �1 � x2� / x2; 1 � x2 � 0 and x � 0; the domain is [�1, 0) � (0, 1].

8. ( f /g)(x) � x3 / �31 � x3�; 1 � x3 � 0, so x � 1; the domain is (��, 1) � (1, �).

( g/f )(x) � �3

1 � x3� / x3; x3 � 0, so x � 0; the domain is (��, 0) � (0, �).

9. 10. 15. f(g(x)) � 3x � 1; (��, �); g( f(x)) � 3x 1; (��, �)

16. f(g(x)) � (�x �

11

�)2� 1 � �

(x �

11)2� � 1; (��, 1) � (1, �); g( f(x)) � �

(x2 �

11) � 1��

x2 �

12

�; (��, ��2� ) � (��2�, �2� ) � (�2�, �)

17. f(g(x)) � x � 1; [�1, �); g( f(x)) � �x2 � 1�; (��, 1] � [1, �) 18. f(g(x)) � ��x�

1

� 1�; [0, 1) � (1, �); g( f (x)) � ; (1, �)

19. f(g(x)) � 1 � x2; [�1, 1]; g( f(x)) � �1 � x4�; [�1, 1] 20. f(g(x)) � 1 � x3; (��, �); g( f(x)) � �3

1 � x9� ; (��, �)

21. f(g(x)) � �32x�; (��, 0) � (0, �); g( f(x)) � �

23x�; (��, 0) � (0, �)

22. f(g(x)) � �1/(x �

11) 1� � �

x �

x1

�; all reals except 0 and 1

g(f(x)) � �1/(x

11) � 1� � ��

x

x1

�; all reals except �1 and 0

23. One possibility: f(x) � �x� and g(x) � x2 � 5x 24. One possibility: f(x) � (x 1)2 and g(x) � x3

25. One possibility: f(x) � �x� and g(x) � 3x � 2 26. One possibility: f(x) � 1/x and g(x) � x3 � 5x 327. One possibility: f(x) � x5 � 2 and g(x) � x � 3 28. One possibility: f(x) � ex and g(x) � sin x29. One possibility: f(x) � cos x and g(x) � �x� 30. One possibility: f(x) � x2 1 and g(x) � tan x

31. V � �43

��r3 � �43

��(48 0.03t)3; 775,734.6 in.3 33. t � 3.63 sec 37. y � �2�5� �� x�2� and y � ��2�5� �� x�2�

38. y � �2�5� �� x� and y � ��2�5� �� x� 39. y � �x2 � 2�5� and y � ��x2� �� 2�5� 40. y � �3�x2� �� 2�5� and y � ��3x2 ��25�41. y � 1� x and y � x � 142. y � x 1 and y � �x � 1 43. y � x and y � �x or y � �x� and y � ��x� 44. y � �x� and y � ��x�

1��x�� 1�

[–5, 5] by [–10, 25][0, 5] by [0, 5]

�x�� 4��x�� 2�

�x�� 2��x�� 4�

x2

��x�� 3�

�x�� 3��

x2



45. False; x is not in the domain of (f/g)(x) if g(x) � 0.46. False; it is the set of all numbers that belong to both the domain of f and the domain of g.51. 55.

SECTION 1.5 Exploration 11. T starts at �4, at the point (�8, �3). It stops at T � 2, at the point (8, 3). 61 points are computed.2. The graph is smoother because the plotted points are closer together.3. The graph is less smooth because the plotted points are further apart.4. The smaller the Tstep, the slower the graphing proceeds. This is because the calculator has to compute more X and Y values. 5. The grapher skips directly from the point (0, �1) to the point (0, 1), corresponding to the T-values T � �2 and T � 0. The two points are connected by a straight line, hidden by the Y-axis. 6. With Tmin set at �1, the grapher begins at the point (�1, 0),missing the bottom side of the curve entirely. 7. Leave everything else the same, but change Tmin back to �4 and Tmax to �1.

Exercises 1.55. (a) (�6, �10), (�4, �7), (�2, �4), (0, �1), (2, 2), 6. (a) (�2, 15), (�1, 8), (0, 3), (1, 0), (2, �1), (3, 0), (4, 3)

(4, 5), (6, 8) (b) x2 � 4x 3; It is a function.(b) 1.5x � 1; It is a function. (c)(c)

7. (a) (9, �5), (4, �4), (1, �3), (0, �2), (1, �1), 8. (a) (0, �5), (1, �3), (�2�, �1), (�3�, 1)(4, 0), (9, 1) (b) y � 2x2 � 5; It is a function.

(b) x � (y 2)2; It is not a function. (c)(c)

13. f �1(x) � �13

�x 2, (��, �) 14. f �1(x) � �12

�x � �25

�, (��, �) 15. f �1(x) � �2x

�

3x

�, (��, 2) � (2, �)

16. f �1(x) � �2xx�

13

�, (��, 1) � (1, �) 17. f �1(x) � x2 3, x � 0 18. f �1(x) � x2 � 2, x � 0

[–1, 5] by [–5, 1][–2, 4] by [–6, 4]

[–5, 5] by [–3, 3][–1, 5] by [–2, 6]

[–9.4, 9.4] by [–6.2, 6.2]


f g D

ex 2 ln x (0, �)

(x2 2)2 �x � 2� [2, �)

(x2 � 2)2 �2 � x� (��, 2]

�(x �

11)2� �

x

x1

� x � 0

x2 � 2x 1 x 1 (��, �)

��x

x1

��2

�x �

11

� x � 1


19. f �1(x) � �3x�, (��, �) 20. f �1(x) � �3

x � 5�, (��, �) 21. f �1(x) � x3 � 5, (��, �) 22. f �1(x) � x3 2, (��, �)

23. One-to-one 24. Not one-to-one 25. One-to-one 26. Not one-to-one

27. f(g(x)) � 3[�13

�(x 2)] � 2 � x 2 � 2 � x; g( f(x)) � �13

�[(3x � 2) 2] � �13

�(3x) � x

28. f(g(x)) � �14

�[(4x � 3) 3] � �14

�(4x) � x; g( f(x)) � 4[�14

�(x 3)] � 3 � x 3 � 3 � x

29. f(g(x)) � [(x � 1)1/3]3 1 � (x � 1)1 1 � x � 1 1 � x; g( f(x)) � [(x3 1) � 1]1/3 � (x3)1/3 � x1 � x

30. f(g(x)) � � �71

� � �7x

� � x; g( f(x)) � � �71

� � �7x

� � x

31. f(g(x)) � � (x � 1)(�x �

11

� 1) � 1 x � 1 � x; g( f(x)) � � ( ) � �xx

� � �x 1

x� x

� � �1x

� � x

32. f(g(x)) � � � � � (�xx �

� 11

�) ��22xx

33

�

32((xx

�

�

11))

�� 55x� � x;

g( f(x)) � � � (�xx �

�

22

�) � � �55x� � x

33. (b) y � �2257�x. This converts euros (x) to dollars (y).

34. (a) c�1(x) � �95

�x 32. This converts Celsius temperature to Fahrenheit temperature.

(b) �59

�x 255.38. This converts Fahrenheit temperature to Kelvin temperature.

35. y � ex and y � ln x are inverses. If we restrict the domain of the function y � x2 to the interval [0, �), then the restricted function and y � �x�are inverses. 36. y � x and y � 1/x are their own inverses. 39. True. All the ordered pairs swap domain and range values. 40. True.This is a parametrization of the line y � 2x 1.45. (Answers may vary.) (a) If the graph of f is unbroken, its reflection in the line y � x will be also. (b) Both f and its inverse must be one-to-one inorder to be inverse functions. (c) Since f is odd, (�x, �y) is on the graph whenever (x, y) is. This implies that (�y, �x) is on the graph of f �1

whenever (x, y) is. That implies that f �1 is odd. (d) Let y � f(x). Since the ratio of y to x is positive, the ratio of x to y is positive. Anyratio of y to x on the graph of f �1 is the same as some ratio of x to y on the graph of f, hence positive. This implies that f �1 is increasing.46. (Answers may vary.) (a) f(x) � ex has a horizontal asymptote; f �1(x) � ln x does not. (b) f(x) � ex has domain all real numbers; f �1(x) � ln x does not. (c) f(x) � ex has a graph that is bounded below; f �1(x) � ln x does not.

(d) f(x) � �xx

2 �

�

255

� has a removable discontinuity at x � 5 because its graph is the line y � x 5 with the point (5, 10) removed.

The inverse function is the line y � x � 5 with the point (10, 5) removed. This function has a removable discontinuity, but not at x � 5.

47. (a) y � 0.75x 31 (b) y � �43

�(x � 31). It converts scaled scores to raw scores.

2(x 3) 3(x � 2)��

x 3 � (x � 2)

2(�xx

�

32

�) 3

��

�xx

�

32

� � 1

2(�xx

�

32

�) 3

��

�xx

�

32

� � 1

�2xx�

13

� 3

��

�2xx�

13

� � 2

�2xx�

13

� 3

��

�2xx�

13

� � 2

1��

�x

x1

� � 1

1��

�x

x1

� � 1

�x �

11

� 1

��

�x �

11

�

7�

�7x

�

7�

�7x

�

x

y

5

3



SECTION 1.6 Exploration 11. They raise or lower the parabola along the y-axis. 2. They move the parabola left and right along the x-axis. 3. Yes

Exploration 21. Graph C. Points with positive y-coordinates remain unchanged, while points with negative y-coordinates are reflected across the x-axis.2. Graph A. Points with positive x-coordinates remain unchanged. Since the new function is even, the graph for negative x values will be a reflection of the graph for positive x values.3. Graph F. The graph will be a reflection across the y-axis of graph C.4. Graph D. The points with negative y-coordinates in graph A are reflected across the y-axis.

Exploration 31. The 1.5 and the 2 stretch the graph

vertically; the 0.5 and the 0.25 shrink the graph vertically.

Exercises 1.61. Vertical translation down 3 units 2. Vertical translation up 5.2 units 3. Horizontal translation left 4 units

4. Horizontal translation right 3 units 5. Horizontal translation to the right 100 units 6. Vertical translation down 100 units

7. Horizontal translation to the right 1 unit, and vertical translation up 3 units

8. Horizontal translation to the left 50 units and vertical translation down 279 units

9. Reflection across x-axis 10. Horizontal translation right 5 units 11. Reflection across y-axis

12. This can be written as y � ��(x �� 3)� or y � ��x �3�. The first of these can be interpreted as reflection across the y-axis followed by ahorizontal translation to the right 3 units. The second may be viewed as a horizontal translation left 3 units followed by a

reflection across the y-axis. 13. Vertically stretch by 2 14. Horizontally shrink by �12

�, or vertically stretch by 23 � 8

15. Horizontally stretch by �01.2� � 5, or vertically shrink by 0.23 � 0.008 16. Vertically shrink by 0.3

17. Translate right 6 units to get g 18. Translate left 4 units, and reflect across the x-axis to get g

19. Translate left 4 units, and reflect across the x-axis to get g 20. Vertically stretch by 2 to get g21. 22. 23. 24.

25. f(x) � �x�� 5� 26. f(x) � ��(x� �� 3�)� � �3� �� x� 27. f(x) � ��x�� 2� 3 � 3 � �x�� 2� 28. f(x) � 2�x�� 5� � 329. (a) �x3 5x2 3x � 2 (b) �x3 � 5x2 3x 2 30. (a) �2�x�� 3� 4 (b) 2�3� �� x� � 4

31. (a) y � �f(x) � �(�3 8x�) � �2�3x� (b) y � f(�x) � �3 8(�x)� � �2�3

x� 32. (a) �3�x 5� (b) 3�5 � x�

y

gh

f

x

10

5–5

–10

y

h

f

g

x

3

6–6

–6

y

g h

f

x

10

3–7

y

f g

h

x

10

6–2

[–4.7, 4.7] by [–1.1, 5.1]


2. The 1.5 and the 2 shrink the graphhorizontally; the 0.5 and the 0.25stretch the graph horizontally.

[–4.7, 4.7] by [–1.1, 5.1]


33. Let f be an odd function; that is, f(�x) � �f(x) for all x in the domain of f. To reflect the graph of y � f(x) across the y-axis, we make thetranformation y � f(�x). But f(�x) � �f(x) for all x in the domain of f, so this transformation results in y � �f(x). That is exactly the translation that reflects the graph of f across the x-axis, so the two reflections yield the same graph.34. Let f be an odd function; that is, f(�x) � �f(x) for all x in the domain of f. To reflect the graph of y � f(x) across the y-axis, we make thetransformation y � f(�x). Then, reflecting across the x-axis yields y � �f(�x). But f(�x) � �f(x) for all x in the domain of f, so we have y ��f(�x) � �[�f(x)] � f(x); that is, the original function.35. 36. 37. 38.

39. (a) 2x3 � 8x (b) 27x3 � 12x 40. (a) 2�x 2� (b) �3x 2� 41. (a) 2x2 2x � 4 (b) 9x2 3x � 2

42. (a) �x

22

� (b) �3x

1 2� 43. Starting with y � x2, translate right 3 units, vertically stretch by 2, and translate down 4 units.

44. Starting with y � �x�, translate left 1 unit, vertically stretch by 3, and reflect across x-axis.

45. Starting with y � x2, horizontally shrink by �31

� and translate down 4 units.

46. Starting with y � �x�, translate left 4 units, vertically stretch by 2, and reflect across x-axis, and translate up 1 unit.51. 52. 53. 54.

55. Reflections have more effect on points that are farther away from the line of reflection. Translations affect the distance of points from theaxes, and hence change the effect of the reflections.56. The x-intercepts are the values at which the function equals zero. The stretching (or shrinking) factors have no effect on the number zero, sothose y-coordinates do not change.

57. First vertically stretch by �95

�, then translate up 32 units. 58. First vertically shrink by �59

�, then translate down �1690

� � 17.7� units.

65. (a) (b) Change the y-value by multiplying by theconversion rate from dollars to yen, a numberthat changes according to international market conditions. This results in a vertical stretch by the conversion rate.

36

35

34

33

32

31

1 82 3 4 5 6 7

Pri

ce (

doll

ars)

Month

y

x

y

x

–5

5

5

y

x

–5

–5

5

5

y

x5

5

–5

y

x–5

–5

5

5

x

y

x

y

x

y

x

y



67. (a) The original graph is on the top; (b) The original graph is on the top; the graph of y � |f(x)| is on the bottom. the graph of y � f(|x|) is on the bottom.

(c) (d)

68. (a) You should get a graph that looks like this: (b) Let x � 2 cos t and leave y unchanged.

(c) Let x � 3 cos t and y � 3 sin t. (d) Let x � 4 cos t and y � 2 sin t.

[–4.7, 4.7] by [–3.1, 3.1][–4.7, 4.7] by [–3.1, 3.1]

[–4.7, 4.7] by [–3.1, 3.1][–4.7, 4.7] by [–3.1, 3.1]

y

x

y

x

[–5, 5] by [–10, 10][–5, 5] by [–10, 10]

[–5, 5] by [–10, 10][–5, 5] by [–10, 10]



Documents

5144 Demana TE Ans pp895-914 Answers Ch... · 51. (d) 52. (b) 53. (a)(b)(c) The year 2006 is represented by x 16. So the value of y for x 16 is 6527 million, a little larger than