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Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The rational function f(x) =400 +
5x
x models the cost, f(x), in dollars, to produce x bobble-head figures. The graph is
show. Use the equation to solve the problem.
1)
Find and interpret f(40).
1)
A)
15; It costs $15 to produce 40 bobble-head figures.
B)
600; It costs $600 to produce 40 bobble-head figures.
C)
5; It costs $5 to produce 40 bobble-head figures.
D)
405; It costs $405 to produce 40 bobble-head figures.
The rational function f(x) =120x
100 - x models the cost, f(x), in millions of dollars, to remove x% of the trash from American
highways. The graph is shown. Use the equation to solve the problem.
2)
Find and interpret f(60).
2)
A)
18; The cost to remove 60% of the trash is $18 .
B)
40; The cost to remove 60% of the trash is $40 million.
C)
18; The cost to remove 60% of the trash is $18 million.
D)
18; The cost to remove 40% of the trash is $18 million.
The rational function f(x) =400 +
5x
x models the cost, f(x), in dollars, to produce x bobble-head figures. The graph is
show. Use the equation to solve the problem.
3)
What is the horizontal asymptote of the graph? What does this mean about the cost to produce x
bobble-heads?
3)
A)
y = 5; As the number of bobble-head figures produced increases, the cost is approaching $5.
B)
y = 400; As the number of bobble-head figures produced increases, the cost is approaching
$400.
C)
y = 0; As the number of bobble-head figures produced increases, the cost is approaching $0
D)
There is no horizontal asymptote.
The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
4)
How does the graph indicate that f(-4) does not exist?
4)
A)
There is a horizontal asymptote at y = - 4 indicated by a dashed line.
B)
There is a vertical asymptote at y = - 4 indicated by a dashed line.
C)
There is a vertical asymptote at x = - 4 indicated by a dashed line.
D)
There is a horizontal asymptote at x = - 4 indicated by a dashed line.
Solve the problem.
5)
An airport limo service charges riders a fixed charge of $19 plus $4.00 per mile. How many miles
must a rider go to have an average cost per mile of $4.80?
5)
A)
23.8 miles
B)
5 miles
C)
47.5 miles
D)
28.8 miles
Solve the rational equation.
6)
1 +1
x - 1 =6
x2- 1
6)
A)
-1
3, 1
2
B)
{-3, 2}
C)
{3, 2}
D)
{3, -2}
4
Divide.
7)
(21x2- 29x - 72) ÷ (7x + 9)
7)
A)
3x - 8
B)
x - 8
C)
21x - 8
D)
-8x + 1
8)
24x4- 24x3+ 30x2
6x3
8)
A)
4x - 4 +5
x
B)
4x - 24x3+5
x
C)
9x - 4
D)
4x - 4
Provide an appropriate response.
9)
One pump can drain a pool in 7 minutes. When a second pump is also used, the pool only takes 5
minutes to drain. How long would it take the second pump to drain the pool if it were the only
pump in use?
9)
A)
211
12 min
B)
33 min
C)
2
35 min
D)
17 1
2 min
Use synthetic division to divide.
10)
(x2- 144) ÷ (x + 12)
10)
A)
x + 12
B)
12x - 12
C)
x - 144
D)
x - 12
5
Solve the problem.
11)
The U.S. Maritime Administration estimated that the cost per ton of building an oil tanker could be
represented by the model y =103,000
x +210 , where y is the cost in dollars per ton and x is the tons (in
thousands). What size of oil tanker (in thousands of tons) can be built for $200 per ton?
11)
A)
251 thousand tons
B)
31 thousand tons
C)
725 thousand tons
D)
305 thousand tons
Find the function value.
12)
f(x) =x - 3
4x + 7 ; f(-2)
12)
A)
1
B)
5
C)
0
D)
-5
Divide.
13)
36x8+ 36x6+ 42x4
6x6
13)
A)
6x2+ 6 +7
x2
B)
6x + 6 +7
x
C)
6x2+ 6 +7
x
D)
6x + 6 +7
x2
Provide an appropriate response.
14)
A boat moves 9 kilometers upstream in the same amount of time it moves 19 kilometers
downstream. If the rate of the current is 8 kilometers per hour, find the rate of the boat in still
water.
14)
A)
17 1
10 km/hr
B)
71
5 km/hr
C)
8 km/hr
D)
22 2
5 km/hr
Write an equation to describe the variation. Use k for the constant of proportionality.
15)
x varies inversely as t
15)
A)
x=t
k
B)
t=x
k
C)
x=k
t
D)
t= kx
Perform the indicated operations. Simplify where possible.
16)
x2
x2-1
·x2+3x +2
x2+2x
16)
A)
x
x -1
B)
x -1
x
C)
x
x -3
D)
x -3
x
Divide as indicated.
17)
x2-12x +36
2x -12 ÷5x -30
10
17)
A)
x2-12x +36
(x -6)2
B)
1
C)
10
D)
(x -6)2
4
If y varies directly as x, find the direct variation equation for the situation.
18)
y =6 when x =12
18)
A)
y =1
2x
B)
y =1
6x
C)
y =2x
D)
y = x +6
Simplify the complex fraction.
19)
-9x-1- 3y-1
-7x-2+ 8y-2
19)
A)
9xy2+ 3x2y
7y2+ 8x2
B)
-7y2+ 8x2
-9xy2- 3x2y
C)
-9xy2- 3x2y
-7y2+ 8x2
D)
-7x2+ 8y2
-9x - 3y
Perform the indicated operations. Simplify where possible.
20)
x2-6x -16
x3+3x2+2x
· (x2- 3x - 4) ÷x2-7x +12
3x
20)
A)
3x -3
x +3
B)
3x -24
x -3
C)
3x -24
x +3
D)
3x +3
x -3
Simplify the rational expression.
21)
2
x-5
x +4
4
x2+4x
21)
A)
6- x
2
B)
4- x
C)
10 - x
2
D)
4- x
2
Provide an appropriate response.
22)
The U.S. Maritime Administration estimated that the cost per ton of building an oil tanker could be
represented by the model f(x) =120,000
x +210 , where f(x) is the cost in dollars per ton and x is the tons
(in thousands). What size of oil tanker (in thousands of tons) can be built for $350 per ton?
22)
A)
13 thousand tons
B)
553 thousand tons
C)
133 thousand tons
D)
214 thousand tons
Divide.
23)
(x2- 64) ÷ (x + 8)
23)
A)
x - 8
B)
x2- 8
C)
x - 64
D)
x + 64
Solve the rational equation.
24)
1
x +7+3
x +4=
-3
x2+11x +28
24)
A)
{-7}
B)
C)
{4}
D)
{0}
Divide as indicated.
25)
x2+ 7x + 12
x2+ 11x + 28
÷x2+ 3x
x2+ 3x - 28
25)
A)
x
x2+ 11x + 28
B)
x - 4
x2+ 7x
C)
x - 4
x
D)
x - 4
Use synthetic division to divide.
26)
(-3x3- 14x2- 10x - 8) ÷ (x + 4)
26)
A)
-3
4x2-7
2x -5
2
B)
-3x + 2
C)
-3x2- 2x - 2
D)
3x2- 4x - 2
9
Simplify the complex fraction.
27)
3
3r - 1 - 3
3
3r - 1 + 3
27)
A)
2 + 3r
3r
B)
2 - 3r
3r
C)
3r
2 - 3r
D)
2 - r
r
Find the least common denominator of the rational expressions.
28)
x - 3
x2+ 3x and -18
x2+ 9x + 18
28)
A)
(x - 3)2
B)
x(x - 3)(x + 6)
C)
x(x - 3)2
D)
x(x + 3)(x + 6)
Solve.
29)
When the temperature stays the same, the volume of a gas is inversely proportional to the pressure
of the gas. If a balloon is filled with 93 cubic inches of a gas at a pressure of 14 pounds per square
inch, find the new pressure of the gas if the volume is decreased to 31 cubic inches.
29)
A)
39 pounds per square inch
B)
31
14 pounds per square inch
C)
28 pounds per square inch
D)
42 pounds per square inch
Use the graph or table to determine a solution of the equation. Use synthetic division to verify that this number is a
solution of the equation. Then solve the polynomial equation.
30)
x3+ 9x2+ 26x + 24 = 0
4
-2 2
-4
[-2, 2, 1] by [-4, 4, 1]
30)
A)
-2; The remainder is zero; {-3, -2, 4}
B)
-2; The remainder is zero; {-4, -2, 3}
C)
-2; The remainder is zero; {-4, -3, 2}
D)
-2; The remainder is zero; {-4, -3, -2}
Solve the problem.
31)
To calculate the drug dosage for a child, a pharmacist may use the formula d(x) =Dx
x +11 , 0
x
12.
The child's age is x and the adult dosage is D. What is the dosage for an 6-year old child if the adult
dosage is 40 mg? (Round to the nearest tenth.)
31)
A)
14.1 mg
B)
21.8 mg
C)
40.0 mg
D)
2.4 mg
Subtract. Simplify the result, if possible.
32)
x
x2- 16
-7
x2+ 5x + 4
32)
A)
x2- 6x + 28
(x - 4)(x + 4)(x + 1)
B)
x2- 6
(x - 4)(x + 4)(x +1)
C)
x2+ 6x + 28
(x - 4)(x + 4)(x + 1)
D)
x2- 6x + 28
(x - 4)(x + 4)
Simplify the rational expression. If the rational expression cannot be simplified, so state.
33)
15x2+21x + 6
3x + 3
33)
A)
5x + 21
3x + 5
B)
5x + 2
C)
Cannot be simplified
D)
5x + 2
3x
Divide.
34)
(3x2+ 31x + 36) ÷ (x + 9)
34)
A)
3x + 4
B)
3x2- 31
C)
3x - 4
D)
x + 31
Simplify the complex fraction.
35)
(a-1+ b-1)-1
35)
A)
ab
a
B)
ab
a - b
C)
a + b
ab
D)
ab
a + b
12
Solve the problem.
36)
Jim can run 5 miles per hour on level ground on a still day. One windy day, he runs 11 miles with
the wind, and in the same amount of time runs 7 miles against the wind. What is the rate of the
wind?
36)
A)
22 1
2 mph
B)
11
9 mph
C)
2 mph
D)
5 mph
Subtract. Simplify the result, if possible.
37)
9x2
x - 1 -9x
x - 1
37)
A)
9x(x + 1)
x - 1
B)
9x
x - 1
C)
9x
D)
0
Solve.
38)
If the resistance in an electrical circuit is held constant, the amount of current flowing through the
circuit is directly proportional to the amount of voltage applied to the circuit. When 4 volts are
applied to a circuit, 100 milliamperes of current flow through the circuit. Find the new current if the
voltage is increased to 9 volts.
38)
A)
250 milliamperes
B)
36 milliamperes
C)
216 milliamperes
D)
225 milliamperes
If y varies directly as x, find the direct variation equation for the situation.
39)
y =20 when x =12
39)
A)
y =4x
B)
y =3
5x
C)
y = x + 8
D)
y =5
3x
The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
40)
Find f(-2).
40)
A)
2
B)
1
C)
-2
D)
0
Find the function value.
41)
f(x) =x3+ 3
x2+ 7 ; f(-5)
41)
A)
-125
32
B)
-122
25
C)
7
8
D)
-61
16
Divide.
42)
(6x5- 12x4+ 16x3) ÷(2x4)
42)
A)
3x - 12x4+8
x
B)
3x - 6
C)
3x - 6 +8
x
D)
11x - 6
14
Solve.
43)
The voltage across a resistor is jointly proportional to the resistance of the resistor and the current
flowing through the resistor. If the voltage across a resistor is 40 volts for a resistor whose
resistance is 5 ohms and when the current flowing through the resistor is 8 amperes, find the
voltage across a resistor whose resistance is 6 ohms and when the current flowing through the
resistor is 9 amperes.
43)
A)
54 volts
B)
72 volts
C)
45 volts
D)
48 volts
44)
If the voltage, V, in an electric circuit is held constant, the current, I, is inversely proportional to the
resistance, R. If the current is 240 milliamperes when the resistance is 2 ohms, find the current
when the resistance is 12 ohms.
44)
A)
80 milliamperes
B)
40 milliamperes
C)
1440 milliamperes
D)
1434 milliamperes
Solve the rational equation.
45)
1 -3
2x =7
4
45)
A)
{-2}
B)
-1
2
C)
{2}
D)
1
2
Use the graph or table to determine a solution of the equation. Use synthetic division to verify that this number is a
solution of the equation. Then solve the polynomial equation.
46)
2x3+ 11x2+ 17x + 6 = 0
X Y 1
-2 0
-1-2
0 6
1 36
2 100
3 210
X = - 2
46)
A)
-2; The remainder is zero; -2, -1
2, 3
B)
-2; The remainder is zero; -3, -2, -1
2
C)
-2; The remainder is zero; -3, -2, 1
2
D)
-2; The remainder is zero; -3, -1
2, 2
Write an equation to describe the variation. Use k for the constant of proportionality.
47)
s varies jointly as t and u.
47)
A)
stu = k
B)
s= k +t+u
C)
s= ktu
D)
s+t+u= k
Solve the problem.
48)
A company has monthly fixed costs of $21,000 for its facilities and it costs $230 per unit for each
unit that it produces. How many units must the company produce to have an average cost per unit
of $400?
48)
A)
91 units
B)
135 units
C)
124 units
D)
125 units
Find the domain of the rational function.
49)
f(x) =x2- 25
x2- 10x + 21
49)
A)
domain of f: ( , -5) (-5, 5) (5, )
B)
domain of f: ( , 0) (0, )
C)
domain of f: ( , -7) (-7, -3) (-3, )
D)
domain of f: ( , 3) (3, 7) (7, )
Divide as indicated.
50)
(y -4)2
6÷6y -24
36
50)
A)
1
y -4
B)
y -4
C)
(y -4)3
36
D)
6(y -4)2
6y -24
Multiply as indicated.
51)
x2-20x +99
x2-6x +5
·x2-13x +40
x2-19x +90
51)
A)
(x -11)
(x -10)
B)
(x2-20x +99)(x2-13x +40)
(x2-6x +5)(x2-19x +90)
C)
(x +11)(x +8)
(x +1)(x +10)
D)
(x -11)(x -8)
(x -1)(x -10)
Solve the rational equation.
52)
5 - a
a+3
4=7
a
52)
A)
{8}
B)
{-4}
C)
29
20
D)
{-8}
Solve.
53)
The amount of paint needed to cover the walls of a room varies jointly as the perimeter of the room
and the height of the wall. If a room with a perimeter of 30 feet and 6-foot walls requires 1.8 quarts
of paint, find the amount of paint needed to cover the walls of a room with a perimeter of 80 feet
and 10-foot walls.
53)
A)
80 quarts
B)
8 quarts
C)
800 quarts
D)
16 quarts
Provide an appropriate response.
54)
Divide using synthetic division: (4x3- 22x2+ 8x + 10) ÷ (x - 5).
54)
A)
4x + 2
B)
-4x2+ 5x - 2
C)
4
5x2-22
5x +8
5
D)
4x2- 2x - 2
Multiply as indicated.
55)
4x3
5·40
x2
55)
A)
x
32
B)
32
x
C)
32x2
x3
D)
32x
Perform the indicated operations. Simplify the result, if possible.
56)
3
x+6
x - 2
56)
A)
6x - 9
x(2- x)
B)
9x - 6
x(x - 2)
C)
6x - 9
x(x - 2)
D)
9x - 6
x(2- x)
57)
6
x2+3x
+7
x+2
x +3
57)
A)
7
x
B)
9
x
C)
2
x
D)
14
x
Simplify the rational expression. If the rational expression cannot be simplified, so state.
58)
2x + 4
3x2+ 8x + 4
58)
A)
Cannot be simplified
B)
2x + 3
3x + 8
C)
2x
3x + 2
D)
2
3x + 2
Perform the indicated operations. Simplify the result, if possible.
59)
y -4
y2-9
+4- y
9-y2
59)
A)
8
(y +3)(y -3)
B)
2y
(y +3)(y -3)
C)
2y -8
(y +3)(y -3)
D)
0
Simplify the rational expression. If the rational expression cannot be simplified, so state.
60)
y2- 5y - 24
y2- 6y - 27
60)
A)
-5y - 24
-6y - 27
B)
Cannot be simplified
C)
y - 8
y - 9
D)
-5y - 8
-6y - 9
Solve the problem.
61)
Find the average of 8
5 and 3
4 (Hint: To find the average of two numbers, find their sum and divide
by 2.)
61)
A)
3
5
B)
11
18
C)
47
40
D)
47
10
Perform the indicated operations. Simplify the result, if possible.
62)
x
x2- 16
-4
x2+ 5x + 4
62)
A)
x2- 3x + 16
(x - 4)(x + 4)
B)
x2- 3x + 16
(x - 4)(x + 4)(x + 1)
C)
x2- 3
(x - 4)(x + 4)(x + 1)
D)
x2+ 3x + 16
(x - 4)(x + 4)(x + 1)
The graph of a rational function, f, is shown in the figure. Use the graph to answer the question.
63)
Is 4 a function value of f?
63)
A)
Yes
B)
No
