MTH 112
Practice Problems for Test 3
Solve the polynomial inequality and graph the solution set on a number line. Express the solution set in interval
notation.
1) (x - 2)(x + 5) > 0
1)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
2) x2 - 2x - 24 ≤ 0
2)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
3) (x + 2)(x - 1)(x - 6) > 0
3)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval
notation.
x-8
4)
4)
<0
x+2
5)
(x - 1)(3 - x)
≤0
(x - 2)2
-12 -10 -8
-6
-4
5)
-2
0
2
4
6
8
10 12
Solve the inequality.
12
10
6)
>
x-5 x+1
6)
Solve the problem.
7) If y varies directly as the square of x, and y = 40 when x = 8, find y when x = 20.
7)
8) 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 3 volts are applied to a circuit, 75 milliamperes of current flow through the circuit.
Find the new current if the voltage is increased to 11 volts.
8)
9) x varies inversely as y2 , and x = 4 when y = 12. Find x when y = 4.
9)
1
Solve.
10) 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 90 milliamperes when the resistance is 4
ohms, find the current when the resistance is 12 ohms.
Write an equation that expresses the relationship. Use k for the constant of proportionality.
11) q varies jointly as r and s and inversely as the square root of a.
Write an equation that expresses the relationship. Use k as the constant of variation.
12) The weight of a body above the surface of the earth is inversely proportional to the square
of its distance from the center of the earth. What is the effect on the weight when the
distance is multiplied by 4?
Find the variation equation for the variation statement.
13) c varies directly as a and inversely as b; c = 4 when a = 48 and b = 48
10)
11)
12)
13)
Solve the problem.
14) f varies jointly as q2 and h, and f = 64 when q = 4 and h = 2. Find f when q = 3 and h = 4.
14)
15) 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 70 feet and 8-foot walls
requires 5.6 quarts of paint, find the amount of paint needed to cover the walls of a room
with a perimeter of 45 feet and 6-foot walls.
15)
Approximate the number using a calculator. Round your answer to three decimal places.
16) 4 2.5
Solve the problem.
17) A city is growing at the rate of 0.6% annually. If there were 4,840,000 residents in the city in
1993, find how many (to the nearest ten-thousand) are living in that city in 2000. Use
y = 4,840,000(2.7)0.006t.
Graph the function by making a table of coordinates.
5 x
18) f(x) =
3
6
y
2
-4
-2
2
4
17)
18)
4
-6
16)
6 x
-2
-4
-6
2
The graph of an exponential function is given. Select the function for the graph from the functions listed.
19)
y
10
5
-10
-5
5
10
x
-5
-10
A) f(x) = 5 x - 1
B) f(x) = 5 x - 1
C) f(x) = 5 x
D) f(x) = 5 x + 1
Approximate the number using a calculator. Round your answer to three decimal places.
20) e1.6
Solve the problem.
21) The size of the bear population at a national park increases at the rate of 4.2% per year. If
the size of the current population is 138, find how many bears there should be in 3 years.
Use the function f(x) = 138e0.042t and round to the nearest whole number.
Use the compound interest formulas A = P 1 +
20)
21)
r nt
and A = Pe rt to solve.
n
22) Find the accumulated value of an investment of $800 at 10% compounded quarterly for 5
years.
22)
23) Find the accumulated value of an investment of $7000 at 7% compounded continuously for
6 years.
23)
Write the equation in its equivalent exponential form.
24) log 9 = 2
3
25) log
b
24)
243 = 5
25)
Write the equation in its equivalent logarithmic form.
26) 6 3 = x
26)
27) 10x = 1000
27)
Evaluate the expression without using a calculator.
28) log 27
3
29) log
28)
1
2 8
29)
3
19)
30) log
31) 6
11
1
30)
log 17
6
31)
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
32) log (7 · 11)
32)
6
33) log (125x)
5
34) log
33)
7
3 5
34)
x
10,000
35) log
35)
36) logn x8
37) ln
9
36)
x
37)
38) logb (yz 4 )
39) log
40) log
38)
x3
3 y8
2
39)
x
4
40)
41) log x + log y
c
c
41)
42) log (x + 7) - log (x - 3)
4
4
42)
43) ln x + 8ln y
43)
44) 3 log x + 5 log (x - 6)
6
6
44)
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
45) log 28
4
46) log
15
40.1
45)
46)
4
Solve the equation by expressing each side as a power of the same base and then equating exponents.
1
47) 3 (6 - 3x) =
47)
27
48) 16x + 9 = 64x - 5
48)
Solve the exponential equation. Express the solution set in terms of natural logarithms.
49) 8 3x = 2.3
50) e3x = 6
49)
50)
Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the
solution.
51) 2 x + 6 = 3
51)
52) e2x - 8 - 10 = 1215
52)
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
53) log x = 5
53)
3
54) ln x = 2
54)
55) log 4 + log x = 1
3
3
55)
56) log 3 (x + 6) + log 3 (x - 6) - log 3 x = 2
56)
57) log (x + 4) = 3 + log (x + 1)
5
5
57)
58) log 4x = log 5 + log (x - 3)
58)
59) The value of a particular investment follows a pattern of exponential growth. In the year
2000, you invested money in a money market account. The value of your investment t
years after 2000 is given by the exponential growth model A = 3300e0.053t. How much did
59)
Solve.
you initially invest in the account?
60) The population of a particular country was 29 million in 1980; in 1989, it was 35 million.
The exponential growth function A =29ekt describes the population of this country t years
after 1980. Use the fact that 9 years after 1980 the population increased by 6 million to find
k to three decimal places.
5
60)
REVIEW FROM PREVIOUS MATERIAL
Find and simplify the difference quotient
f(x + h) - f(x)
, h≠ 0 for the given function.
h
61) f(x) = 5x2
61)
Evaluate the piecewise function at the given value of the independent variable.
x2 - 3
if x ≠ 7
62) h(x) = x - 7
; h(7)
x-8
62)
if x = 7
Graph the function.
x+5
63) f(x) = -4
-x + 5
if -8 ≤ x < 2
if x = 2
if x > 2
63)
y
10
5
-10
-5
5
10
x
-5
-10
Begin by graphing the standard square root function f(x) =
function.
64) g(x) = - x + 1 + 2
10
8
x . Then use transformations of this graph to graph the given
64)
y
6
4
2
-10 -8 -6 -4 -2-2
2
4
6 8 10 x
-4
-6
-8
-10
Find the inverse of the one-to-one function.
5
65) f(x) =
7x - 8
65)
6
Find the domain of the function.
66) f(x) = 6 - x
66)
For the given functions f and g , find the indicated composition.
x-4
67) f(x) =
,
g(x) = 5x + 4
5
67)
(g∘f)(x)
Find a rational zero of the polynomial function and use it to find all the zeros of the function.
68) f(x) = 2x3 - 9x2 + 7x + 6
68)
Solve the polynomial equation. In order to obtain the first root, use synthetic division to test the possible rational roots.
69) 4x3 - 23x2 + 26x + 8 = 0
69)
Find an nth degree polynomial function with real coefficients satisfying the given conditions.
1
70) n = 4; 3, , and 1 + 2i are zeros; f(1) = 48
3
7
70)
Answer Key
Testname: TEST 3 PRACTICE PROBLEMS SPR 11
1) (-∞, -5) ∪ (2, ∞)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
2) [-4, 6]
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
3) (-2, 1) ∪ (6, ∞)
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
4) (-2, 8)
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
5) (-∞, 1] ∪ [3, ∞)
-12 -10 -8
-6
-4
-2
0
2
4
6
8
10 12
6) (-31, -1) or (5, ∞)
7) 250
8) 275 milliamperes
9) x = 36
10) 30 milliamperes
krs
11) q =
a
12) The weight is divided by 16
4a
13) c =
b
14) f = 72
15) 2.7 quarts
16) 32.000
17) 5,050,000
18)
6
y
4
2
-6
-4
-2
2
4
6 x
-2
-4
-6
19) C
20) 4.953
21) 157
8
Answer Key
Testname: TEST 3 PRACTICE PROBLEMS SPR 11
22) $1310.89
23) $10,653.73
24) 3 2 = 9
25) b5 = 243
26) log x = 3
6
27) log
1000 = x
10
28) 3
29) -3
30) 0
31) 17
32) log 7 + log 11
6
6
33) 3 + log x
5
34) log 7 - log 5
3
3
35) log x - 4
36) 8logn x
37)
1
ln x
9
38) logb y + 4 logb z
39) 3 log x - 8 log y
3
3
1
40) log x - 2
2
2
41) log (xy)
c
x+7
42) log
4 x-3
43) ln xy8
44) log x3 (x - 6)5
6
45) 2.4037
46) 1.3631
47) {3}
48) 33
ln 2.3
49)
3 ln 8
50)
ln 6
3
51) -4.42
52) 7.56
53) {243}
54) e2
3
55) { }
4
56) {12}
121
57) 124
58) {15}
59) $3300.00
9
Answer Key
Testname: TEST 3 PRACTICE PROBLEMS SPR 11
60) 0.021
61) 5(2x+h)
62) -1
63)
y
10
(2, 7)
5
-10
(-8, -3)
(2, 3)
-5
5
10
x
-5 (2, -4)
-10
64)
10
y
8
6
4
2
-10 -8 -6 -4 -2-2
2
4
6 8 10 x
-4
-6
-8
-10
5
8
65) f-1 (x) =
+
7x 7
66) (-∞, 6]
67) x
1
68) - , 2, 3
2
69) -
1
, 2, 4
4
70) f(x) = -6x4 + 48x3 - 114x2 + 168x - 45
10