Math 1101 Exam 3 Practice Problems
These problems are not intended to cover all possible test topics. These problems should serve
as an activity in preparing for your test, but other study is required to fully prepare. These problems
contain some multiple choice questions, please consult with your instructor for particular details
about your class test.
1. Decide whether or not the functions
f (x) =
5
,
x+4
g(x) =
4x + 5
x
are inverses of each other.
Solution:
4x + 5
f (g(x)) = f
x
x
5
= 4x+5
+4 x
x
5x
=
4x + 5 + 4x
5x
=
8x + 5
Since the composite is not equal to x, they are not inverses of each other! You may also
check the graphs of each function to determine if they are reflections over the line y = x.
2. Determine whether or not the function is one-to-one.
(a)
A. Yes
B. No
MATH 1101
Exam 3 Review, Page 2 of 12
Fall 2013
(b) f (x) = 8x2 − 3
A. Yes B. No
3. Determine whether the function that pairs students’ ID numbers with their GPAs is an invertible function.
A. Yes
B. No
4. Find the inverse of the function
f (x) =
5x − 2
2x − 8
Solution:
5x − 2
2x − 8
5y − 2
x=
2y − 8
x(2y − 8) = 5y − 2
2xy − 8x = 5y − 2
2xy − 8x − 5y = −2
2xy − 5y = 8x − 2
y(2x − 5) = 8x − 2
8x − 2
y=
2x − 5
8x − 2
f −1 (x) =
2x − 5
y=
5. Graph the function using transformations of an exponential function’s graph with which you
are familiar
f (x) = 4x−2 + 3
Solution: The graph of f (x) is obtained by shifting the graph of g(x) = 4x horizontally
2 units to the right and vertically 3 units upward.
MATH 1101
Exam 3 Review, Page 3 of 12
Fall 2013
6. A computer is purchased for $4800. Its value each year is about 78% of the value the preceding year. Its value, in dollars, after t years is given by the exponential function V (t) =
4800(0.78)t . Find the value of the computer after 7 years.
Solution: V (7) = 4800(0.78)7 = 843.15, so the value is $843.15 after 7 years.
7. What are the domain and range for the equation y = 2x ?
A. Domain: (0, ∞); Range (−∞, ∞)
B. Domain: (−∞, ∞); Range (0, ∞)
C. Domain: (−∞, ∞); Range (−∞, ∞)
D. Domain: (−∞, ∞); Range [0, ∞)
8. Write the logarithmic equation in exponential form.
(a) log5 125 = 3
Solution: 53 = 125
(b) ln x = −7
A. x = e7
B. x = ln −7
C. No solution
9. Find the value of
log6
without using a calculator.
A. 6
B. −2
C. 2
D. −6
1
36
D. x = e−7
MATH 1101
Exam 3 Review, Page 4 of 12
Fall 2013
10. Graph the function f (x) = log2 (x) + 4.
Solution: The graph of f (x) is obtained by shifting the graph of g(x) = log2 x vertically
4 units upward.
11. Find the inverse of the function f (x) = 8x .
A. f −1 (x) = logx 8
B. f −1 (x) = ln 8x
C. f −1 (x) = log8 x
D. f −1 (x) = x8
12. Solve the equation log5 x = −2
1
1
C. −10 D.
A. 3 B.
32
25
13. A certain noisehasan intensity I of 8.17×10−5 . Give that decibel level L is related to intensity
by L = 10 log II0 , where I0 = 10−12 , determine the decibel level of the noise. Round your
answer to the nearest decibel.
A. 69 decibels
B. 182 decibels
C. 79 decibels
D. 8 decibels
14. Rewrite the expression as the sum and/or difference of logarithms, without using exponents.
Simplify if possible.
√
3
11
log16 2
y x
MATH 1101
Exam 3 Review, Page 5 of 12
Fall 2013
Solution:
√
3
log16
√
11
3
=
log
11 − log16 (y 2 x)
16
y2x
1
= log16 11 − (log16 y 2 + log16 x)
3
1
= log16 11 − (2 log16 y + log16 x)
3
1
= log16 11 − 2 log16 y − log16 x
3
15. Rewrite as a single logarithm.
1
1
1
log2 x4 + log2 x4 − log2 x
2
4
6
A. log2 x17/6
B. log2 x9/2
C.
7
6
log2 x8
D. log2 x7
16. Solve the equation by hand. Give the exact answer and then an approximation rounded to the
nearest thousandth.
55x−1 = 23
Solution:
55x−1 = 23
log5 55x−1 = log5 23
(5x − 1) log5 5 = log5 23
5x − 1 = log5 23
5x = log5 23 + 1
log5 23 + 1
x=
5
x ≈ 0.5896
17. Solve the equation. Give an exact solution.
log(x − 3) = 1 − log x
A. 5
B. −5, 2
C. −5
D. −2, 5
MATH 1101
Exam 3 Review, Page 6 of 12
Fall 2013
18. The sales of a new model of notebook computer are approximated by S(x) = 4000−14000e−x/9 ,
where x represents the number of months the computer has been on the market, and S represents the sales in thousands of dollars. In how many months will the sales reach $15,000?
Solution: We would like to find the solution to 4000 − 14000e−x/9 = 1500, one can do
that in Graph by plotting both the left and right hand sides of the equation simultaneously
and finding the intersection of the resulting graphs:
This gives an answer of 15.5 months.
19. Assume the cost of a car is $27,000. With continuous compounding in effect, the cost of the
car will increase according to the equation C = 27000ert , where r is the annual inflation rate
and t is the number of years. Find the number of years it would take to double the cost of the
car at an annual inflation rate of 5.2%. Round the answer to the nearest hundredth.
A. 209.55 years
B. 1.96 years
C. 13.33 years
D. 196.22 years
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Exam 3 Review, Page 7 of 12
Fall 2013
20. Find the exponential function f that models this data. Round the coefficients to the nearest
hundredth.
x 1
y 580
2
620
3
670
4
750
A. f (x) = (1.09)(527.34)x
B. f (x) = (567.57)(0.17)x
C. f (x) = (527.34)(1.09)x
D. f (x) = (0.17)(567.57)x
21. Find the exponential function which models a solid with initial mass of 22 grams, decreasing
at a rate of 3.1% per day.
Solution: An exponential function which models growth or decay rate of r percent with
an initial value of P0 is given by P (t) = P0 (1 + r)t . In this case, r = −0.031 and P0 = 22
the function is:
P (t) = 22(0.969)t
22. Find an exponential function that models the data below and use it to predict about how many
books will have been read in the eighth grade.
Grade
2
3
4
5
A. 330
B. 4496
C. 1883
Number of Books Read
9
27
67
121
D. 788
MATH 1101
Exam 3 Review, Page 8 of 12
Fall 2013
23. Several years ago, a large city undertook a major effort to encourage carpooling in order
to reduce traffic congestion. The accompanying table shows the number of carpoolers, in
thousands, from 1995 to 2000. Use regression to obtain a function f (x) = a + b ln x that
models the data, where x = 1 corresponds to 1995, x = 2 to 1996, and so on. Round the
constants a and b to the nearest hundredth.
1995
Year
Carpoolers 3.9
1996
8.1
1997
10.3
1998
11.8
1999
12.6
2000
13.2
A. f (x) = 4.40 + 5.32 ln x
B. f (x) = 4.33 + 5.17 ln x
C. f (x) = 4.22 + 5.25 ln x
D. f (x) = 4.29 + 5.14 ln x
24. Select an appropriate type of modeling function for the data shown in the graph.
y
x
A. Linear
B. Cubic
C. Exponential
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Exam 3 Review, Page 9 of 12
Fall 2013
25. Does it appear that a linear model or an exponential model is the better fit for the data given in
the table below?
x
y
2
5
4 7.5
6 9.8
8 12.3
Solution:
x
y
2
5
4 7.5
6 9.8
8 12.3
1st diff.
2.5
2.3
2.5
% change
50%
30.7%
25.5%
Since the first differences are relatively constant, and the percent change is not, a linear
model is the better fit for this data.
26. Jorge invested $2500 at 4% interest compounded semi-annually. In how many years will
Jorge’s investment have tripled? Round your answer to the nearest tenth of a year.
Solution: If P0 dollars are invested at a rate of r compounded k times a year, the value
of the investment in t years is given by V (t) = P0 (1 + kr )kt . In this case, P0 = 2500,
r = 0.04, and k = 2 giving the function
2t
0.04
V (t) = 2500 1 +
= 2500(1.02)2t
2
Jorge’s investment triples once its value becomes $7500:
7500 = 2500(1.02)2t
3 = (1.02)2t
log 3 = log(1.02)2t
log 3 = 2t log 1.02
log 3
= 2t
log 1.02
log 3
=t
2 log 1.02
In this case, t = 27.7 years.
MATH 1101
Exam 3 Review, Page 10 of 12
Fall 2013
27. State the degree of the polynomial and whether the leading coefficient is positive or negative.
A. 4: Leading coefficient is negative.
B. 5: Leading coefficient is negative.
C. 4: Leading coefficient is positive.
D. 5: Leading coefficient is positive.
28. State the degree and leading coefficient of the polynomial
f (x) = −2(x + 9)2 (x − 9)2
A. Degree 4; leading coefficient 1
B. Degree 4; leading coefficient −2
C. Degree 2; leading coefficient −2
D. Degree 2; leading coefficient 1
29. Predict the end behavior of the graph of the function
f (x) = −1.48x4 − x3 + x2 − 8x + 4
A. Down on both sides
B. Up on both sides
C. Up on left side, down on right side
D. Down on left side, up on right side
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Exam 3 Review, Page 11 of 12
Fall 2013
30. P (x) = −x3 + 27
x2 − 60x + 100, x ≥ 5 is an approximation to the total profit (in thousands
2
of dollars) from the sale of x hundred thousand tires. Find the number of tires that must be
sold to maximize profit.
A. 5.5 hundred thousand
B. 4 hundred thousand
C. 5 hundred thousand
D. 4.5 hundred thousand
31. Approximate the coordinates of each turning point accurate to two decimal places
y = x4 − 4x3 + 4x + 6
Solution: To approximate the coordinates of the turning points, you should plot the function in Graph.
Then you may use the trace to “Snap to” the extreme y-values. These yield local minima
of (−0.5321, 4.5544) and (2.8794, −9.2344) and a local maximum of (0.6527, 7.6800).
MATH 1101
Exam 3 Review, Page 12 of 12
Fall 2013
32. The table below gives the number of births, in thousands, to females over the age of 35 for a
particular state every two years from 1970 to 1986
Year
1970
1972
1974
1976
1978
1980
1982
1984
1986
Births (thousands)
42.5
29.9
36.0
56.9
71.1
69.9
57.2
37.1
25.9
Use technology to find the quartic function that is the best fit for this data, where x is the
number of years after 1970. According to the model, how many births were to females over
the age of 35 in this state in 1990?
A. 108,868
B. 106,368
C. 101,318
D. 94,368