Name: Class: Date: 1. f x

f ( x ) =6 if x 27x 6 if x > 2{

f ( x ) = x2

+ 4x if x 67x if x > 6{

f ( a + h) andf ( a + h) f ( a )

hwhere h 0.

1. Express the rule: "Divide by 15, then subtract 6" in function notation

and the function f.

[For example the rule: "Square, then subtract 5" is expressed as the function f (x) =(x2 5) .]

2. Express the rule: "Square, add 20, then take the square root" in function notation

and the function f.

[For example: the rule "square, then subtract 5" is expressed as the function f (x) = (x2 5).]

3. Evaluate the following piecewise defined function at the values f ( 10 ), f ( 2 ), and f ( 10 ).

4. Evaluate the following piecewise defined function at the values f ( 4 ), f ( 6 ), and f ( 9 ).

5. Use the function f (x) = x2 + 9 to evaluate the following expressions and simplify.

f ( x + 2 ) and f ( x ) + f ( 2 )

6. Use the function f (x) = 4 x + 2 to evaluate the following expressions and simplify the result.

f (x2) and (f (x))2

7. For the function f (x) = 3 x2 + 5 find the following:

PAGE 1

Name: __________________ Class: Date: _____________

f ( a + h) f ( a )h

where h 0.

f ( x ) = 6x + 15, 4 x 6

f ( x ) =1

4x 12

f ( x ) =x + 6

x2

4

f ( t ) =3

3t 4

G ( x ) = 1 x2

g ( x ) =x

x2

+ 3x 18

8. For function f (x) = 5 x3 find

9. Find the domain of the following function:






PAGE 2

Name: __________________ Class: Date: _____________

g ( x ) =8

x2

13x


16. Graphs of the functions f and g are given. For which values of x is f (x) = g (x)?

PAGE 3

Name: __________________ Class: Date: _____________

f (x) = (4 x2)

17. Find the domain and range of the function that is graphed below.

18. Find the domain of the function .

PAGE 4

Name: __________________ Class: Date: _____________

19. Find a function of the line segment joining the points (8, 13) and (9, 14).

20. Write an equation that expresses that G varies directly as x.

21. Write an equation that expresses that F is directly proportional to z.

22. Write an equation that expresses that z varies inversely as G.

23. Write an equation that expresses that F is proportional to the square root of t.

24. Write an equation that expresses that N is proportional to the square of t and inversely proportional to the cube of B.

25. Write an equation that expresses that N is jointly proportional to the squares of z and A.

26. Write an equation that expresses that G is proportional to m and inversely proportional to d and z.

27. Express the statement "y is directly proportional to x" as a formula. Use the information that if x = 18 then y = 90 to find theconstant of proportionality.

28. Express the statement "M varies directly as x and inversely as y" as a formula. Use the information that if x = 8 and y = 4 thenM = 10 to find the constant of proportionality.

29. The pressure P of a sample of gas is directly proportional to the temperature T and inversely proportional to the volume V. Findthe constant of proportionality if 100 L of gas exerts a pressure of 15.37 kPa at a temperature of 290 K (absolute temperaturemeasured on the Kelvin scale). If the temperature is increased to 600 K and the volume is decreased to 50 L, what is the pressureof the gas?

PAGE 5

Name: __________________ Class: Date: _____________

f (x) =1x

f (x) = x

30. The resistance R of a wire varies directly as its length L and inversely as the square of its diameter d. Find the constant ofproportionality if a wire 46.8 m long and 0.006 m in diameter has a resistance of 78 ohms. Find the resistance of a wire made ofthe same material that is 1 m long and has a diameter of 0.002 m.

31. The cost of a sheet of gold foil is proportional to its area. If a rectangular sheet measuring 72 cm by 64 cm costs $921.6, howmuch would a 9 cm by 8 cm sheet cost?

32. In the short growing season of the Canadian arctic territory of Nunavut, some gardeners find it possible to grow giganticcabbages in the midnight sun. Assume that the final size of a cabbage is proportional to the amount of nutrients it receives, andinversely proportional to the number of other cabbages surrounding it. A cabbage that received 35 oz of nutrients and had 6other cabbages around it grew to 33 lb. What size would it grow to if it received 7 oz of nutrients and had only 3 cabbage"neighbors"?

33. The value of a building lot on Galiano Island is jointly proportional to its area and the quantity of water produced by a well onthe property. A 200 ft by 400 ft lot has a well producing 8 gallons of water per minute, and is valued at $20000. What is thevalue of a 800 ft by 800 ft lot if the well on the lot produces 4 gallons of water per minute?

34. Determine the average rate of change of the function f (x) = 6 2 x between x = 2 and x = 3.

35. Determine the average rate of change of the function f (t) = t2 6 t between t = 2 and t = 1.

36. Determine the average rate of change of the function f (x) = x3 7 x2 between x = 0 and x = 8.

37. Determine the average rate of change of the function f (x) = x + x 2 between the x = 0 and x = 2.

38. Determine the average rate of change of the function between the x = 1 and x = 8.

39. Determine the average rate of change of the function between x = 4 and x = 9.

PAGE 6

Name: __________________ Class: Date: _____________

f (x) =13

x + 640. Find the average rate of change of the function: between x = a and x = a + h.

41. The graph of a function is given below. Determine the average rate of change of the function between the indicated values of thevariable.

PAGE 7

Name: __________________ Class: Date: _____________


PAGE 8

Name: __________________ Class: Date: _____________


44. A man is running around a circular track 200 m in circumference. An observer uses a stopwatch to time each lap, obtaining thedata in the following table. What was the man’s average speed (rate) between 84 s and 200 s?

Time (s) Distance (m)

38 200

84 400

138 600

200 800

270 1000

348 1200

434 1400

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Name: __________________ Class: Date: _____________

45. The table shows the number of CD players sold in a small electronics store in the years 1989 1999. What was the average rateof change of sales between 1989 and 1999?

Year CD players sold

1989 510

1990 710

1991 535

1992 735

1993 600

1994 715

1995 665

1996 535

1997 580

1998 580

1999 630

46. Determine the interval on which the function in the graph below is decreasing.

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Name: __________________ Class: Date: _____________

47. The graph of a function is given below. Determine the interval on which the function is decreasing.

48. If we reflect the graph of h (x) in the y axis, we will obtain the graph of what function?

PAGE 11

Name: __________________ Class: Date: _____________

48. The graph of g (x) = x 2 is given :

The function u (x) was obtained g (x) and is graphed below. What is u (x)?

49. The function f (x) is reflected in the x axis and then shifted up 6 units and the graph of g (x) = 6 x 2 is obtained. What is f (x)?

50. Fill in the blank to make the following statement true.To obtain the graph of g(x) = x 6, we reflect the graph of f (x) = x 6 in the ___ axis.

PAGE 12

Name: __________________ Class: Date: _____________

50. The graph of g(x) = 8 |x| is obtained by shifting f (x) up 8 units. What is f (x)?

50. The graphs of f( r) and u( r) are shown in the following illustration (blue and red respectively):

If f (r) =( r2 2) what is u( r)?

50. The graph of the function y = x2 + 6 x is:

Find the coordinates of its vertex.

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Name: __________________ Class: Date: _____________

51. The graph of the function y = x2 8 x + 7 is:

Find the coordinates of its x intercepts.

52. The graph of the function y = x2 + 5 x is:

Find its maximum or minimum value.

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Name: __________________ Class: Date: _____________

53. The graph of the function y = x2 2 x + 7 is:

Express the quadratic function in standard form.

54. Find the maximum or minimum value of the function f (x) = x2 + x + 9.

55. Find the maximum value of the function f (x) = 4 x2 + 40 x 60.

56. Find the minimum value of the function f (x) = 8 x2 80 x.

57. Find a function whose graph is a parabola with vertex (4, 60) and that passes through the point (1, 24).

58. Find the domain and range of the function f (x) = x2 2 x + 5.

PAGE 15

Name: __________________ Class: Date: _____________

U (x) = x 8 x

V (x) =5 x

2

x3

59. A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function

R (x) = 396 x 0.9 x2

where the revenue R (x) is measured in dollars. What is the maximum revenue, and how many units should be manufactured toobtain this maximum?

60. Find the minimum value of the quadratic function f (x) = x2 + 3.64 x + 7.2 , correct to two decimal places.

61. Find the local maximum value of the function f (x) = 5 x3 2 x and the value of x at which it occurs. State each answer correct totwo decimal places.

62. Find the local minimum values of the function g (x) = x4 4 x3 4 x2 and the value of x at which each occurs. State each answercorrect to two decimal places.

63. Find the local maximum value of the function and the value of x at which it occurs. State each answercorrect to two decimal places.

64. Find the local minimum value of the function and the value of x at which it occurs. State each answer correct

to two decimal places.

65. Find the maximum value of the function f (x) = x4 + 4 x2 + 5. (Hint: Let t = x2.)

66. A rectangular building lot is four times as long as it is wide. Find a function that models its area ,A ,in terms of its width ,w.

67. The height of a cylinder is two times its radius. Find a function that models the volume V of the cylinder in terms of its radius r,V(r).

PAGE 16

Name: __________________ Class: Date: _____________

68. A rectangle has a perimeter of 20 ft. Find a function that models its area A in terms of the length x of one of its sides, A(x).

69. A rectangle has an area of 12 m 2. Find a function, P(x), that models its perimeter P in terms of the length x of one of its sides.

70. A rectangular box with a volume of 90 ft 3 has a square base. Find a function, S(x) that models its surface area S in terms of thelength x of one side of its base.

71. A woman 7 ft tall is standing near a street lamp that is 13 ft tall, as shown in the figure. Find a function, L(d), that models thelength L of her shadow in terms of her distance d from the base of the lamp.

a = 7, b = 13

72. Two ships leave port at the same time. One sails south at 18 mi/h and the other sails east at 24 mi/h. Find a function that modelsthe distance D between the ships in terms of the time t ( in hours ) elapsed since their departure.

73. The sum of two positive numbers is 80. Find a function, P(x), that models their product P in terms of x, one of the numbers.

74. An isosceles triangle has a perimeter of 32 cm. Find a function, A(b), that models its area A in terms of the length of its base b.

PAGE 17

Name: __________________ Class: Date: _____________

75. A rectangle is inscribed in a semicircle of radius 50, as shown in the figure. Find a function that models the area A of therectangle in terms of its height h.

76. Find two numbers whose sum is 18 and whose product is a maximum.

PAGE 18

Name: __________________ Class: Date: _____________

h = (x + 5) 5x 30

f (x) =x + 7x 7

f (x) = 3x 1 g(x) = 9 x2

(g f )( 5)

77. Find the dimensions that give the largest area for the rectangle shown in the figure. Its base is on the x axis and its other twovertices are above the x axis, lying on the parabola y = 6 x 2. Round your answer to one decimal place.

78. Find the domain of the function

79. Find the domain of the function

80. Use and to evaluate the expression .

PAGE 19

Name: __________________ Class: Date: _____________

( f g)(2)

g f , if f (x) = x2

and g(x) = x 16

f g, if f (x) =7

x 3 and g(x) =8

x 3

f (x) = x2

+ 3, g (x) = x 10, and h (x) = x f g h.

F (x) = (x 10)5

f g

81. Use the given graphs of f and g to evaluate the expression .

82. Find the domain of .

83. f (x) = x 4 and g(x) = |x + 2|. Find g (f (x)).

84. Find the domain of .

85. For find

86. Express the function in the form .

PAGE 20

Name: __________________ Class: Date: _____________

F (x) =8

x + 1f g

F(x) = (2 x )5

f g h

A A



89. An airplane is flying at a speed of 300 mph at an altitude of h miles. The plane passes directly above a radar station at time t = 0.

Express the distance (in miles) between the plane and the radar station as a function of time t (in hours) that the plane has flownif h = 2.

90. A savings account earns 5% interest compounded annually. If you invest x dollars in such an account, then the amount A(x) ofthe investment after one year is the initial investment plus 5%; that is,

A(x) = x + 0.05 x = 1.05 x

represents the amount of the investment after two years. Find a formula for what you get when you compose 6 copies ofA.

91. The graphs of the functions u(x) = m1x + b1 and w(x) = m2x + b2 are lines with slopes m1 and m2 respectively. What is the slopeof the graph of u( w(x))?

PAGE 21

Name: __________________ Class: Date: _____________

f (x) =1

x + 5

x 8x 5

f (x) =1 8x6 3x

f (x) = 10x + 11

f (x) = 3 +3

x

92. Suppose that g(x) = 3 x + 2 and h(x) = 9 x 2 + 12 x + 11. Find a function f, such that f(g(x)) = h(x). (Think about what operationsyou would have to perform on the formula for g to end up with the formula for h.)

93. Assume g is a one to one function. If g(x) = x2 + 6 x with x 3, find g 1 (10)

94. Find the inverse function of f (x) = 3 x + 12.

95. Find the inverse function of .



98. Find the inverse function of f (x) = 1 2 x3.



101. Find the inverse function of f (x) = (9 x3) 1/5

PAGE 22

Name: __________________ Class: Date: _____________

f (x) = 2 + x + 8

f ( x ) = 9 x2

, 3 x 3

f (x) = x5

+ 9

f (x) = (x + 1)2



104. Find the inverse function of

105. The function is not one to one.

a. What is the least restrictive domain so that the resulting function is one to one?b. Find the inverse of the function with the restricted domain.

PAGE 23

Name: __________________ Class: Date: _____________

f x( )=x15

6

f x( )= x2+20

6,6,640,60,63

f x+2( )=x2+4x+13,f x( )+f 2( )=x

2+22

f x2( )=4x

2+2, f x( )( )

2=16x

2+16x+4

f a+h( )=3h2+3a

2+6h a+5,

f a+h( ) f a( )h

=3h+6a

f a+h( ) f a( )h

=15a2+15a h+5h

2

4,6x ,3( ) 3,( )x , 2( ) 2,2( ) 2,( )

,( )1,1

3,( ),0( 13, )

14,5 , 2,22,2

f x( )=x+5G=c xF=c z

z=cG

F=c t

N=c t

2

B3

N=c z2

A2

G=c md z

555.3,63.60.00006,15$14.413.2$80000.0000000000000000000029

83

18151364

1.

2. 3. 4.

5.

6.

7.

8.

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

22.

23.

24.

25.

26.

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

38.

39.

40.

41.

PAGE 1

ANSWER KEY

Name: __________________ Class: Date: _____________

124

93.4512

9,9( )2,0( )

u x( )=x2+3

f x( )=x2

3,9( )7,0( ) , 1,0( )2.5,6.25( )

y= x 1( )2+6

f12

=8.75

f 5( )=40f 5( )= 200

4x2

32x+4,( ) , 4, )

43560,220f 1.82( )=3.89f 0.37( )=0.49g 0.56( )= 0.45,g 3.56( )= 70.55U 5.33( )=8.71V 3.87( )= 0.179

A w( )=4w2,w>0

V r( )=2 r3,r>0

A x( )=10x x2,0<x<10

P x( )=2x+24x

,x>0

S x( )=2x2+

360x

,x>0

L d( )=76

d,d>0

D t( )=30t,t 0

P x( )=80x x2,80>x>0

A b( )=b 64 4b,0<b<16

A h( )=2h 2500 h2

,0<h<509, 9

2.8,4.06, )7,7) 7,( )

2474

, 4( 4, )g f x( )( )= x 23, )

F x( )= x 10( ) 2+3

f x( )=x5,g x( )=x 10

42.

43. 44. 45. 46. 47.

48.

49. 50. 51. 52.

53.

54.

55. 56.

57. 58. 59. 60. 61. 62. 63. 64. 65.

66.

67.

68.

69.

70.

71.

72.

73. 74.

75. 76. 77. 78. 79. 80. 81. 82. 83. 84.

85.

86.

PAGE 2

ANSWER KEY

Name: __________________ Class: Date: _____________

f x( )=8x

,g x( )=x+1

f x( )=x5,g x( )=2 x,h x( )= x

4+90000t2

1.34xm

1m

2

f=x2+7

3+ 19x 12( )

3

5+1x

8 5x( )1 x( )6x 1( )3x 8( )

3 x 1( )2

x2

11( )10

x 3( )3

39 x

5

x 2( )2

8

9 x2

5x 91, ), x 1

87.

88.

89. 90. 91.

92. 93.

94.

95.

96.

97.

98.

99.

100.

101.

102.

103.

104. 105.

PAGE 3

ANSWER KEY

Name: __________________ Class: Date: _____________

Name: Class: Date: 1. f x

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