Practice Test - Chapter 3
Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end
behavior, and where the function is increasing or decreasing.
1. f (x) = –ex + 7
SOLUTION: Evaluate the function for several x-values in its domain.
x
−7
−5 −4
−3
y
−1 −7.4 −20.1 −54.6
Then use a smooth curve to connect each of these ordered pairs.
List the domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.
7
D = (– , ); R = (– , 0); y-intercept: –e ; asymptote: x-axis;
decreasing for (–
,
)
2. SOLUTION: Evaluate the function for several x-values in its domain.
x
3
4
0
2
−4
−1
y −3.74 −2.8 −2 1.56
5.26
11.43
Then use a smooth curve to connect each of these ordered pairs.
List the domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.
D = (– , ); R = (–4, ); y-intercept: –2, x-intercept: 1.36; asymptote: y = –4;
increasing for (– , )
Use the graph of f (x) to describe the transformation that results in the graph of g(x). Then sketch the
graphs of f (x) and g(x).
3. f (x) = (0.5)x; g(x) = (0.5)x − 3 + 4
SOLUTION: This function is of the form
eSolutions Manual - Powered by Cognero
. Therefore, the graph of g(x) is the graph of f (x) translated 3 units to the
Page 1
right and 4 units up. The translation to the right is indicated by the subtraction of 3 in the exponent. The translation
up
is indicated by the addition of 4.
List the domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.
D = (– , ); R = (–4, ); y-intercept: –2, x-intercept: 1.36; asymptote: y = –4;
Practice Test - Chapter 3 increasing for (– , )
Use the graph of f (x) to describe the transformation that results in the graph of g(x). Then sketch the
graphs of f (x) and g(x).
3. f (x) = (0.5)x; g(x) = (0.5)x − 3 + 4
SOLUTION: This function is of the form
. Therefore, the graph of g(x) is the graph of f (x) translated 3 units to the
right and 4 units up. The translation to the right is indicated by the subtraction of 3 in the exponent. The translation up
is indicated by the addition of 4.
Use the graphs of the functions to confirm this transformation.
4. f (x) = 5x,
SOLUTION: x
This function is of the form f (x) = 5 . Therefore, the graph of g(x) is the graph of f (x) reflected in the x-axis,
reflected in the y-axis, and translated 2 units down. The reflection in the x-axis is indicated by the negative
coefficient in front of the 5. The reflection in the y-axis is indicated by the negative exponent. The translation down
is indicated by the subtraction of 2.
Use the graphs of the functions to confirm this transformation.
5. MULTIPLE CHOICE For which function is A f (x) = –2 · 3−x
B
C f (x) = –log8 (x – 5)
D f (x) = log3 (–x) – 6
eSolutions Manual - Powered by Cognero
Page 2
SOLUTION: For option A, as x approaches infinity, 3
−x
approaches 0, so the function approaches 0. For option B, as x
Practice Test - Chapter 3
5. MULTIPLE CHOICE For which function is A f (x) = –2 · 3−x
B
C f (x) = –log8 (x – 5)
D f (x) = log3 (–x) – 6
SOLUTION: For option A, as x approaches infinity, 3
approaches infinity,
−x
approaches 0, so the function approaches 0. For option B, as x
approaches 0, so the function approaches 0. For option D, as x approaches infinity, log3
(−x) is undefined. For option C, as x approaches infinity, log8 (x − 5) approaches infinity and − log8 (x − 5)
approaches negative infinity. The correct choice is C.
Evaluate each expression.
6. log3
SOLUTION: 7. log32 2
SOLUTION: 8. log 1012
SOLUTION: 9. SOLUTION: eSolutions Manual - Powered by Cognero
Sketch the graph of each function.
Page 3
8. log 10
SOLUTION: Practice Test - Chapter 3
9. SOLUTION: Sketch the graph of each function.
10. f (x) = –log4 (x + 3)
SOLUTION: Convert the base of the equation.
Evaluate the function for several x-values in its domain.
x
1
2
0
−3
−2 −1
y undef. 0 −0.5 −0.79
−1
−1.16
Then use a smooth curve to connect each of these ordered pairs.
11. g(x) = log (–x) + 5
SOLUTION: Evaluate the function for several x-values in its domain.
x
0
−3
−2 −1
y
5.48 5.30 5 undef.
Then use a smooth curve to connect each of these ordered pairs.
12. FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8% for 12 years, making no
other deposits or withdrawals.
a. What will be your account balance if the interest is compounded monthly?
b. What will be your account balance if the interest is compounded continuously?
c. If your investment is compounded daily, about how long will it take for it to be worth double the initial amount?
SOLUTION: eSolutions
Manual - Powered by Cognero
a.
Page 4
Practice Test - Chapter 3
12. FINANCIAL LITERACY You invest $1500 in an account with an interest rate of 8% for 12 years, making no
other deposits or withdrawals.
a. What will be your account balance if the interest is compounded monthly?
b. What will be your account balance if the interest is compounded continuously?
c. If your investment is compounded daily, about how long will it take for it to be worth double the initial amount?
SOLUTION: a.
b.
c.
Expand each expression.
13. log6 36xy2
SOLUTION: 14. log3
SOLUTION: eSolutions Manual - Powered by Cognero
15. GEOLOGY Richter scale magnitude of an earthquake can be calculated using
energy produced and E0 is a constant.
Page 5
where E is the
Practice Test - Chapter 3
15. GEOLOGY Richter scale magnitude of an earthquake can be calculated using
where E is the
energy produced and E0 is a constant.
a. An earthquake with a magnitude of 7.1 hit San Francisco in 1989. Find the scale of an earthquake that produces
10 times the energy of the 1989 earthquake.
b. In 1906, San Francisco had an earthquake registering 8.25. How many times as much energy did the 1906
earthquake produce as the 1989 earthquake?
SOLUTION: a. If E is the energy produced by the 1989 earthquake, then 10 times this energy is 10E. Use the 1989 data to solve
for E0, then replace E with 10E to find the magnitude of an earthquake with 10 times the energy.
Now, solve for R.
The magnitude is about 7.8.
b. For this problem, we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E. Let the
1906 earthquake = nE.
eSolutions Manual - Powered by Cognero
Page 6
The magnitude is about 7.8.
Practice
Test - Chapter 3
b. For this problem, we need to find the energy of the 1906 earthquake in relation to the 1989 earthquake E. Let the
1906 earthquake = nE.
The 1906 earthquake produced about 53 times as much energy as the 1989 earthquake.
Condense each expression.
16. 2 log4 m + 6 log4 n – 3(log4 3 + log4 j )
SOLUTION: 17. 1 + ln 3 – 4 ln x
SOLUTION: Solve each equation.
18. 3x + 8 = 92x
SOLUTION: eSolutions Manual - Powered by Cognero
Page 7
Practice Test - Chapter 3
Solve each equation.
18. 3x + 8 = 92x
SOLUTION: 19. e2x – 3ex + 2 = 0
SOLUTION: x = ln 2 or 0
20. log x + log (x – 3) = 1
SOLUTION: The logarithm of a negative has no real solution and 2 − 3 = −1. Thus, x = 5.
21. log2 (x – 1) + 1 = log2 (x + 3)
SOLUTION: eSolutions Manual - Powered by Cognero
Page 8
Practice Test - Chapter 3
The logarithm of a negative has no real solution and 2 − 3 = −1. Thus, x = 5.
21. log2 (x – 1) + 1 = log2 (x + 3)
SOLUTION: 22. MULTIPLE CHOICE Which equation has no solution?
F ex = e–x
x–1
x+1
G2
=3
H log5 x = log9 x
J log2 (x + 1) = log2 x
SOLUTION: For option F, x can equal 0. Option G can be solved using the natural log. For option H, x can equal 1. For option J, x
+ 1 must equal x, so it has no solution.
For Exercises 23 and 24, complete each step.
a. Find an exponential or logarithmic function to model the data.
b. Find the value of each model at x = 20.
23. SOLUTION: a. An exponential regression cannot be completed with negative y-values. Calculate the logarithmic regression.
The rounded logarithmic regression equation is f (x) = 8.20 – 5.11ln x.
b. Using the full regression equation, f (20) ≈ −7.11.
eSolutions Manual - Powered by Cognero
24. Page 9
Practice
Test - Chapter 3
24. SOLUTION: a. The exponential regression has the stronger correlation coefficient.
x
The rounded exponential regression equation is f (x) = 2.9(1.1) .
Using the full exponential regression equation, f (20) ≈ 21.22.
25. CENSUS The table gives the U.S. population between 1790 and 1940. Let 1780 = 0.
a. Linearize the data, assuming a quadratic model. Graph the data and write an equation for a line of best fit.
b. Use the linear model to find a model for the original data. Is a quadratic model a good representation of population
growth? Explain.
SOLUTION: a. What linearizing data according to a quadratic model, use (x,
).
Plot the data.
eSolutions Manual - Powered by Cognero
Page 10
Practice Test - Chapter 3
Plot the data.
Find the regression.
The rounded regression equation is y = 0.07x.
b. Substitute
for and solve for y.
2
The rounded regression equation is y = 0.0041x + 0.1331x + 1.0816. Population growth generally does not grow
without bounds. Typically, it will start to level off and plateau. For these reasons, a logistic model would be more
appropriate.
eSolutions Manual - Powered by Cognero
Page 11