Name: __________________
Class:
Date: _____________
1. Express the rule: "Divide by 15, then subtract 6" in function notation
and the function f.
2
[For example the rule: "Square, then subtract 5" is expressed as the function f (x) =(x 5) .]
2. Express the rule: "Square, add 20, then take the square root" in function notation
and the function f.
2
[For example: the rule "square, then subtract 5" is expressed as the function f (x) = (x 5).]
3. Evaluate the following piecewise defined function at the values f ( 10 ), f ( 2 ), and f ( 10 ).
f (x)=
{
6
7x 6
if x 2
if x > 2
4. Evaluate the following piecewise defined function at the values f ( 4 ), f ( 6 ), and f ( 9 ).
f (x)=
{
2
x + 4x
7x
if x 6
if x > 6
2
5. Use the function f (x) = x + 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.
2
2
f (x ) and (f (x))
2
7. For the function f (x) = 3 x + 5 find the following:
f ( a + h) and
PAGE 1
f ( a + h) h
f (a)
where h 0.
Name: __________________
3
8. For function f (x) = 5 x find
Class:
f ( a + h) h
Date: _____________
f (a)
where h 0.
9. Find the domain of the following function:
f ( x ) = 6x + 15,
4
x
6
10. Find the domain of the following function:
1
4x 12
f (x)=
11. Find the domain of the following function:
x + 6
f (x)=
x
2
4
12. Find the domain of the following function:
f (t )=
3
3t 4
13. Find the domain of the following function:
1 x
G(x)=
2
14. Find the domain of the following function:
x
g(x)=
x
PAGE 2
2
+ 3x 18
Name: __________________
Class:
Date: _____________
15. Find the domain of the following function:
8
g(x)=
x
2
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: _____________
17. Find the domain and range of the function that is graphed below.
18. Find the domain of the function f (x) = PAGE 4
2
(4 x ) .
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 the
constant 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 then
M = 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. Find
the constant of proportionality if 100 L of gas exerts a pressure of 15.37 kPa at a temperature of 290 K (absolute temperature
measured on the Kelvin scale). If the temperature is increased to 600 K and the volume is decreased to 50 L, what is the pressure
of the gas?
PAGE 5
Name: __________________
Class:
Date: _____________
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 of
proportionality 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 of
the 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, how
much 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 gigantic
cabbages in the midnight sun. Assume that the final size of a cabbage is proportional to the amount of nutrients it receives, and
inversely proportional to the number of other cabbages surrounding it. A cabbage that received 35 oz of nutrients and had 6
other 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 on
the 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 the
value 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.
2
35. Determine the average rate of change of the function f (t) = t 6 t between t = 2 and t = 1.
3
2
36. Determine the average rate of change of the function f (x) = x 7 x between x = 0 and x = 8.
2
37. Determine the average rate of change of the function f (x) = x + x between the x = 0 and x = 2.
38. Determine the average rate of change of the function f (x) =
39. Determine the average rate of change of the function f (x) =
PAGE 6
1
between the x = 1 and x = 8.
x
x between x = 4 and x = 9.
Name: __________________
Class:
40. Find the average rate of change of the function: f (x) =
Date: _____________
1
x + 6 between x = a and x = a + h.
3
41. The graph of a function is given below. Determine the average rate of change of the function between the indicated values of the
variable.
PAGE 7
Name: __________________
Class:
Date: _____________
42. The graph of a function is given below. Determine the average rate of change of the function between the indicated values of the
variable.
PAGE 8
Name: __________________
Class:
Date: _____________
43. The graph of a function is given below. Determine the average rate of change of the function between the indicated values of the
variable.
44. A man is running around a circular track 200 m in circumference. An observer uses a stopwatch to time each lap, obtaining the
data 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
PAGE 9
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 rate
of 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.
PAGE 10
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: _____________
2
48. The graph of g (x) = x is given :
The function u (x) was obtained g (x) and is graphed below. What is u (x)?
2
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 is obtained. What is f (x)?
50. Fill in the blank to make the following statement true.
6
6
To obtain the graph of g(x) = x , we reflect the graph of f (x) = x 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):
2
If f (r) =( r 2) what is u( r)?
2
50. The graph of the function y = x + 6 x is:
Find the coordinates of its vertex.
PAGE 13
Name: __________________
Class:
2
51. The graph of the function y = x 8 x + 7 is:
Find the coordinates of its x intercepts.
2
52. The graph of the function y = x + 5 x is:
Find its maximum or minimum value.
PAGE 14
Date: _____________
Name: __________________
Class:
Date: _____________
2
53. The graph of the function y = x 2 x + 7 is:
Express the quadratic function in standard form.
2
54. Find the maximum or minimum value of the function f (x) = x + x + 9.
2
55. Find the maximum value of the function f (x) = 4 x + 40 x 60.
2
56. Find the minimum value of the function f (x) = 8 x 80 x.
57. Find a function whose graph is a parabola with vertex (4, 60) and that passes through the point (1, 24).
2
58. Find the domain and range of the function f (x) = x 2 x + 5.
PAGE 15
Name: __________________
Class:
Date: _____________
59. A manufacturer finds that the revenue generated by selling x units of a certain commodity is given by the function
2
R (x) = 396 x 0.9 x
where the revenue R (x) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to
obtain this maximum?
2
60. Find the minimum value of the quadratic function f (x) = x + 3.64 x + 7.2 , correct to two decimal places.
3
61. Find the local maximum value of the function f (x) = 5 x 2 x and the value of x at which it occurs. State each answer correct to
two decimal places.
4
3
2
62. Find the local minimum values of the function g (x) = x 4 x 4 x and the value of x at which each occurs. State each answer
correct to two decimal places.
63. Find the local maximum value of the function U (x) = x 8 x and the value of x at which it occurs. State each answer
correct to two decimal places.
2
64. Find the local minimum value of the function V (x) =
5 x
x
3
and the value of x at which it occurs. State each answer correct
to two decimal places.
4
2
2
65. Find the maximum value of the function f (x) = x + 4 x + 5. (Hint: Let t = x .)
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).
2
69. A rectangle has an area of 12 m . Find a function, P(x), that models its perimeter P in terms of the length x of one of its sides.
3
70. A rectangular box with a volume of 90 ft has a square base. Find a function, S(x) that models its surface area S in terms of the
length 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 the
length 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 models
the 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 the
rectangle 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: _____________
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 two
2
vertices are above the x axis, lying on the parabola y = 6 x . Round your answer to one decimal place.
78. Find the domain of the function h = (x + 5) 5x 30
79. Find the domain of the function
f (x) =
2
x + 7
x 7
80. Use f (x) = 3 x 1 and g(x) = 9 x to evaluate the expression (g PAGE 19
f )( 5) .
Name: __________________
Class:
Date: _____________
81. Use the given graphs of f and g to evaluate the expression ( f 82. Find the domain of g f , if
2
f (x) = x
and g(x) =
g)(2) .
x 16 .
83. f (x) = x 4 and g(x) = |x + 2|. Find g (f (x)).
84. Find the domain of f g, if
f (x) =
7
2
x 3
and g(x) =
85. For f (x) = x + 3, g (x) = x 10, and h (x) =
5
x find f 86. Express the function F (x) = (x 10) in the form f g .
PAGE 20
8
x 3.
g h.
Name: __________________
87. Express the function F (x) =
Class:
Date: _____________
8
in the form f g .
x + 1
88. Express the function F(x) = (2 x)
5
in the form f g h.
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 flown
if 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) of
the investment after one year is the initial investment plus 5%; that is,
A(x) = x + 0.05 x = 1.05 x
A A represents the amount of the investment after two years. Find a formula for what you get when you compose 6 copies of
A.
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 slope
of the graph of u( w(x))?
PAGE 21
Name: __________________
Class:
Date: _____________
2
92. Suppose that g(x) = 3 x + 2 and h(x) = 9 x + 12 x + 11. Find a function f, such that f(g(x)) = h(x). (Think about what operations
you would have to perform on the formula for g to end up with the formula for h.)
2
93. Assume g is a one to one function. If g(x) = x + 6 x with x 94. Find the inverse function of f (x) = 3 x + 12.
95. Find the inverse function of f (x) =
96. Find the inverse function of
1
.
x + 5
x 8
.
x 5
97. Find the inverse function of f (x) =
1 8x
.
6 3x
3
98. Find the inverse function of f (x) = 1 2 x .
99. Find the inverse function of f (x) =
10x + 11 .
100. Find the inverse function of f (x) = 3 +
3
x .
3 1/5
101. Find the inverse function of f (x) = (9 x )
PAGE 22
1
3, find g (10)
Name: __________________
Class:
102. Find the inverse function of f (x) = 2 +
103. Find the inverse function of f ( x ) =
Date: _____________
x + 8 .
2
9 x , 3 x 3.
104. Find the inverse function of
f (x) = x
5
+ 9
2
105. The function f (x) = (x + 1) 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
ANSWER KEY
Name: __________________
Class:
x
6
15
1. f ( x ) =
2
2. f ( x ) = x +20
3. 6,6,64
4. 0,60,63
2
2
5. f ( x+2 ) =x +4x+13,f ( x ) +f ( 2 ) =x +22
( 2)
2
2
2
6. f x =4x +2, ( f ( x ) ) =16x +16x+4
f ( a+h ) f ( a )
2
2
7. f ( a+h ) =3h +3a +6h a+5,
=3h+6a
h
f ( a+h ) f ( a )
2
2
8.
=15a +15a h+5h
h
9. 4,6
10. x ( ,3 ) ( 3,
)
11. x ( , 2 ) ( 2,2 ) ( 2,
)
12. ( ,
)
13. 1,1
14. ( 3,
)
15. ( ,0 13,
)
16. 1
17. 4,5 , 2,2
18. 2,2
19. f ( x ) =x+5
20. G=c x
21. F=c z
c
22. z=
G
23. F=c t
2
24. N=
c t
3
B
2
2
25. N=c z A
c m
26. G=
d z
27. 5
28. 5
29. 5.3,63.6
30. 0.00006,15
31. $14.4
32. 13.2
33. $80000.00000000000000000000
34. 2
35. 9
36. 8
37. 3
1
38.
8
1
39.
5
1
40.
3
6
41.
4
PAGE 1
Date: _____________
ANSWER KEY
Name: __________________
42.
43.
44.
45.
46.
47.
12
4
9
3.45
12
( 9,9 )
( 2,0 )
2
48. u ( x ) =x +3
49.
50.
51.
52.
2
f ( x ) =x
( 3,9 )
( 7,0 ) , ( 1,0 )
( 2.5,6.25)
2
53. y= ( x 1 ) +6
1
54. f
=8.75
2
55. f ( 5) =40
56. f ( 5) = 200
57.
58.
59.
60.
61.
62.
63.
64.
65.
2
4x 32x+4
( ,
) , 4,
)
43560,220
f ( 1.82 ) =3.89
f ( 0.37) =0.49
g ( 0.56 ) = 0.45,g ( 3.56 ) = 70.55
U ( 5.33) =8.71
V ( 3.87) = 0.17
9
2
66. A ( w ) =4w ,w>0
3
67. V ( r ) =2 r ,r>0
2
68. A ( x ) =10x x ,0<x<10
24
69. P ( x ) =2x+
,x>0
x
2 360
70. S ( x ) =2x +
,x>0
x
7
71. L ( d ) = d,d>0
6
72. D ( t ) =30t,t 0
2
73. P ( x ) =80x x ,80>x>0
74. A ( b) =b 64 4b,0<b<16
75.
76.
77.
78.
79.
80.
81.
82.
83.
84.
2
A ( h ) =2h 2500 h ,0<h<50
9, 9
2.8,4.0
6,
)
7,7) ( 7,
)
247
4
( , 4 4,
)
g ( f ( x ) ) = x 2
3,
)
2
85. F ( x ) = ( x 10 ) +3
5
86. f ( x ) =x ,g ( x ) =x 10
PAGE 2
Class:
Date: _____________
ANSWER KEY
Name: __________________
87. f ( x ) =
8
,g ( x ) =x+1
x
5
88. f ( x ) =x ,g ( x ) =2 x,h ( x ) = x
2
89. 4+90000t
90. 1.34x
91. m m
1
2
2
92. f=x +7
93. 3+ 19
( x 12 )
94.
3
1
95. 5+
x
( 8 5x )
96.
( 1 x )
( 6x 1 )
97.
( 3x 8 )
( x 1 )
3
98. 2
99.
( x2 11)
10
3
100. ( x 3 )
3
9 x
101.
5
2
102. ( x 2 ) 8
2
9 x
103.
5
104. x 9
105. 1,
) , x 1
PAGE 3
Class:
Date: _____________