Chapter P Section P. I (page 8)

I. b 2. d 3. a 4. c

9. Answers will vary.

X

y

- 5

3

- 4

2

- 3

1

- 2

0

- 1

1

0

2

1

3

11. Answers will vary. X

y

0

- 4

1

- 3

4

- 2

9

- 1

16

0

; 13. Answers will vary. X

y

- 3 2 3

- 2

- 1

- 1

- 2

0

Undef.

1

2

2

1

3 2 3

15. Xmin = -3 Xmax = 5 Xsc l= l Ymin = -3 Ymax = 5 Yscl = 1

17. y = V 5 - ^

M-00.3) . C U . 7 3 )

. \

35. Symmetric with respect to the origin 37. Symmetric with respect to the y-axis

5.

X

y

Answers will - 4

- 5

- 2

- 2

vary

0

1

2

4

4

7

7. Answers will X

y

- 3

- 5

- 2

0

vary 0

4

2

0

3

- 5

39. y = - 3 x + 2 Symmetry: none

41. y = 5 * - 4 Symmetry: none

43. y = 1 - x2

Symmetry: y-axis 45. y = (x + 3)z

Symmetry: none

47. y = x3 + 2 Symmetry: none

49. y = xVx + 2 Symmetry: none

51. x = y3

Symmetry: origin 53. y = 1/x

Symmetry: origin

55. y = 6 - |x| Symmetry: y-axis

(6.0) <—i x i> i

57. y, = Vx + 9 y 2 = - 7 x + 9 Symmetry: x-axis

(-9.0) (0,3)

(0,-3)

3 £•

(a) y = 1.73 (b) x = -4.00 19. (0. - 2 ) , (-2,0) . (1, 0) 21. (0.0), (5, 0), ( - 5 . 0) 23.(4,0) 25.(0,0) 27. Symmetric with respect to the y-axis 29. Symmetric with respect to the x-axis 31. Symmetric with respect to the origin 33. No symmetry

59. y,

Symmetry: x-axis ~3

61.(1.1) 63. ( -1 ,5) , (2, 2) 65. ( - 1 , - 2 ) . (2,1) 67. ( - 1 , - 1 ) , (0,0), (1,1) 69. ( - 1 , - 5 ) , (0 , -1) , (2,1) 71. ( - 2 , 2), ( - 3 , 7 5 )

75. x » 3133 units 77. y = (x + 2)(x - 4)(x - 6) 79. (i) b; Jfc = 2 (ii) d; Jfc = - 1 0 (iii) a; k = 3 (iv) c; k 81. False. ( - 1 , - 2 ) is not a point on the graph of x = jy2.-83. True 85. x2 + (y - 4)2 = 4

= 36

Section P.2 (page 16) 39. x - 5 = 0 41. 22x - 4y + 3 = 0

II. m is undefined.

(2,5)

(11)

13. m = 2

H—I—I—l-w

IS. (0, 1), (1,1), (3,1) 17. (0, 10), (2, 4), (3,1) 19. (a) J (b) lOVlOft

21. (a) g 290

E 280

(b) Population increased-jlga rapidly: 2000-2001 w S

c -I—I—I—I—1-»-6 1 t 9 10 II Year (6 (-» 1996)

23. m j , (0, 4) 25. m is undefined, no y-intercept 27. 3x - 4y + 12 = 0 29. 2x - 3y = 0

31. 3 x - 11 = 0 33. 3x - y

37. 8x + 3y - 40 = 0

H, 8)

(5.1)

43. 49.

x - 3 = 0 45. 3x + 2y - 6 = 0 47. x + y - 3 = 0 51.

i i i i

53.

57. (a)

The lines in (a) do not appear perpendicular, but they do in (b) because a square setting is used. The lines are perpendicular.

59. (a) 2x - y - 3 = 0 (b) x + 2y - 4 = 0 61. (a) 40x - 24y - 9 = 0 (b) 24x + 40y - 53 = 0 63. (a) x - 2 = 0 (b) y - 5 = 0 65. V = 125r + 2040 67. V.= -2000r + 28,400 69. y = 2x 71. Not collinear, because m, ^ m7

77. 5F - 9C - 160 = 0; 72°F = 22.2*C 79. (a) W, = 12.50 + 0.75x; Wt = 9.20 + 1.30x

(b) *! _, (c) When six units are produced, wages are $17.00 per hour with either option. Choose position 1 when less than six units are produced and position 2 otherwise.

(a) x = (1330 - p)/15 (b) (c) 49 units

^

*>

2 s 1 si

x(655) = 45 units

•ection P.3 (page 27) '• (a) Domain of / : [ -4 ,4}; Range of / : [ -3 ,5]

Domain of g: [- 3, 3]; Range of g: [ -4 ,4] ( b ) / ( - 2 ) = - l ; g ( 3 ) = - 4

; (c) x = - \ (d) x = 1 (e) x = - 1 , x « 1, and x = 2 2- (a) - 3 (b) - 9 (c) 2b - 3 (d) 2x - 5

S. (a) 3 (b) 0 (c) - 1 (d) 2 + 2r - ;2

7. (a) 1 (b) 0 (c) - 5 9. 3X2 + 3xAx + (Ax)2, Ax * 0 11. ( V F ^ T - x + l)/[(x - 2)(x - 1)]

- - l / [ V x - l(l + Vx - i)J x * 2 13. Domain: [-3,00); Range: (— 00,0] 15. Domain: Ail real numbers t such that r i4 4n + 2, where n is an

integer, Range: (—00, — l] U [l, co) 17. Domain: (-00, 0) U (0,00); Range: ( -co, 0) U (0, 00) 19. Domain: [0, l ] 21. Domain: AJ1 real numbers x such that x # 2nir, where n is

an integer. 23. Domain: ( - 0 0 , - 3 ) u ( - 3 , co) 25-. (a) - 1 (b) 2 (c) 6 (d) 2r2 + 4

Domain: ( -co, 00); Range: (—00,1) u [2, co) 27. (a) 4 (b) 0 (c) - 2 (d) -fc2

Domain: (— 00, co); Range: (— co, 0] U [l, co) 2 9 . / ( x ) = 4 - x 31. h(x) = v / x i r T

Domain: (-90,00) Domain: [1, co) Range: (— co, co) Range: [0, co)

1 1 3

33. f{x) = J9=* Domain: [—3, 3] Range: [0,3]

35. g{t) = 2 sin irt Domain: (—00, co) Range: [ -2 ,2]

37. The student travels jnu/niin during the first 4 min, is stationary for the next 2 min, and travels 1 mi/min during the final 4 min

39. y is not a function of x. 41. y is a function of x. 43. y is not a function of x. 45. y is not a function of x. 47. d 48. b 49. c 50. a 51. e 52. g 53. (a) J (b)

doT^f- ~hyp o$~ AOL4- ce>iuu-]i

55. (a) Vertical translation (b) Reflection about the x-axis

i 1 1 1—-z

(c) Horizontal translation

57. (a) 0 (b) 0 (c) - 1 (d) -715 (e) Jx* - 1 (f) x - 1 (x > 0)

59. (f°g)(x) = x; Domain: [0, 00) (g • / ) W = |*fc Domain: (— co, 00) No, their domains are different.

61. (/•*)(*) = 3 / ( x 2 - l ) ; Domain: ( - c o , - l ) u ( - l , l ) u ( l . co ) 6 •/)(*) = 9/x2 ~ 1; Domain: ( -co , 0) U (0, co) No

63. (a) 4 (b) - 2 (c) Undefined. The graph of g does not exist at x = - 5 . (d) 3 (e) 2 (f) Underfined. The graph of/ does not exist at x = - 4 . -

65. Answers will vary. Example: /(x) = Jx\ g(x) = x - 2; h(x) - Ix

67. Even 69. Odd 71. (a) (5,4) (b) (5 , -4) 73. / is even, g is neither even nor odd. h is odd. 75. f[x) = - 2 x - 5 77. y = - J^i 79. if, c = - 2 80. i, c = £ 81. :v, c = 32 82. Hi, c = 3 83. (a) 7X4) - 16°, 7tl5) - 24e

(b) The changes in temperature will occur 1 hr later. (c) The temperatures are 1° lower.

85. (a) (b) A(IS) « 345 acres/fam « • y = jx — 5 or 3x - 2y - 10 = 0

27. y = - § x - 2 or 2x + 3y + 6 = 0

10 JO 10 40 50 Year (0«-»1950)

[2x - 2, if X > 2 87. f{x) = |x| + |x - 2| = J 2, if 0 < x < 2

[ -2x + 2,ifx < 0 89. Proof 91. Proof 93. (a) V[x) = x(24 - 2x)2,x > 0 (b) 4 x 16 x 16 cm

29. (a) 7x - 16y + 78 = 0 (b) 5x - 3y + 22 = 0 (c) 2x + y = 0 (d) x + 2 = 0 V = 12,500 - 850r, $9950 Not a function 35. Function

O T3 "~" o

E

P £

(c) Height, x

1 2 3 4 5 6

Length and Width

24 - 2(1) 24 - 2(2) 24 - 2(3) 24 - 2(4) 24 - 2(5) 24 - 2(6)

Volume, V

37. 1[24 - 2(1)]2 = 484 2[24 - 2(2)P = 800 3[24 - 2(3)? = 972 4[24 - 2(4)? = 1024 5[24-2(5)]2 = 980 { 6[24 - 2(6)p = 864

The dimensions of the box that yield a maximum volume are 4 x 16 x 16 cm.

95. False. For example, if /(x) = x2, then/(-1) = / ( l ) . 97. True 99. Putnam Problem Al, 1988

Review Exercises for Chapter P (page 37)

'• (I. 0), (0, - 3 ) 3. (1, 0), (0, \) 5. y-axis symmetry 7.

H 1 1 1 1 r-»-

•3 * 2 J= - °

£ 5

a o ii I ^ 2 e T3 O = at - a S- E •3 <

^

(a) Undefined (b) - 1/(1 + Ax), Ax * 0, - 1 (a) Domain: [ -6 , 6]; Range: [0,6] (b) Domain: ( -oo, 5) U (5, oo); Range: ( -co , 0) U (0, oo) (c) Domain: (—00,00); Range: (—00,00)

41. (a)

(c) '

43. (a)

/

r All the graphs pass through the origin. The graphs of the odd powers of x are symmetric with respect to the origin and the graphs of the even powers are symmetric with respect to the y-axis. As the powers increase, the graphs become flatter in the interval — 1 < x < 1. Graphs of these equations with odd powers pass through Quadrants I and HI. Graphs of these equations with even powers pass through Quadrants I and H

(b) The graph of y = x7 should pass through the origin and Quadrants I and m. It should be symmetric with respect to the origin and be fairly flat in the interval (— 1', 1). The graph of y = x8 should pass through the origin and Quadrants I and JJ. It should be symmetric with respect to the y-axis and be fairly flat in the interval ( -1 ,1 ) .

a o. a, ft a a c c o o o o 'u '0 '5 "o S E E S O <D 0> D O O O O O O O O 00 cto bo so S E .E -E

'•B " 3 " o T J

J -J CO CO

35 CI •* tS Ml 8 8 8 § | 8> g> SP

•S TI 13 fl E E § 2 E I .1 .1 s 5 *S -E § i s s CO J3 O T3