Chapter 10 Review Exercises
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 18828
Chapter Review Exercises
Finding Composite and Inverse Functions
In the following exercises, for each pair of functions, find
 \((f \circ g)(x)\)
 \((g \circ f)(x)\)
 \((f \cdot g)(x)\)
1. \(f(x)=7 x2\) and \(g(x)=5 x+1\)
2. \(f(x)=4 x\) and \(g(x)=x^{2}+3 x\)
 Answer

2.
 \(4 x^{2}+12 x\)
 \(16 x^{2}+12 x\)
 \(4 x^{3}+12 x^{2}\)
In the following exercises, evaluate the composition.
 For functions \(f(x)=3 x^{2}+2\) and \(g(x)=4 x3\), find
 \((f \circ g)(3)\)
 \((g \circ f)(2)\)
 \((f \circ f)(1)\)
 For functions \(f(x)=2 x^{3}+5\) and \(g(x)=3 x^{2}7\), find
 \((f \circ g)(1)\)
 \((g \circ f)(2)\)
 \((g \circ g)(1)\)
 Answer

2.
 \(123\)
 \(356\)
 \(41\)
In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function onetoone.
 \(\begin{array}{l}{\{(3,5),(2,4),(1,3),(0,2)} , {(1,1),(2,0),(3,1) \}}\end{array}\)
 \(\begin{array}{l}{\{(3,0),(2,2),(1,0),(0,1)} , {(1,2),(2,1),(3,1) \}}\end{array}\)
 \(\begin{array}{l}{\{(3,3),(2,1),(1,1),(0,3)} , {(1,5),(2,4),(3,2) \}}\end{array}\)
 Answer

2. Function; not onetoone
In the following exercises, determine whether each graph is the graph of a function and if so, is it onetoone.

Figure 10.E.1
Figure 10.E.2

Figure 10.E.3
Figure 10.E.4
 Answer

1.
 Function; not onetoone
 Not a function
In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.
 \(\{(3,10),(2,5),(1,2),(0,1)\}\)
 Answer

1. Inverse function: \(\{(10,3),(5,2),(2,1),(1,0)\}\). Domain: \(\{1,2,5,10\}\). Range: \(\{3,2,1,0\}\).
In the following exercise, graph the inverse of the onetoone function shown.
 Answer

Solve on your own
In the following exercises, verify that the functions are inverse functions.
 \(\begin{array}{l}{f(x)=3 x+7 \text { and }} {g(x)=\frac{x7}{3}}\end{array}\)
 \(\begin{array}{l}{f(x)=2 x+9 \text { and }} {g(x)=\frac{x+9}{2}}\end{array}\)
 Answer

1. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.
 \(f(x)=6 x11\)
 \(f(x)=x^{3}+13\)
 \(f(x)=\frac{1}{x+5}\)
 \(f(x)=\sqrt[5]{x1}\)
 Answer

1. \(f^{1}(x)=\frac{x+11}{6}\)
3. \(f^{1}(x)=\frac{1}{x}5\)
Evaluate and Graph Exponential Functions
In the following exercises, graph each of the following functions.
 \(f(x)=4^{x}\)
 \(f(x)=\left(\frac{1}{5}\right)^{x}\)
 \(g(x)=(0.75)^{x}\)
 \(g(x)=3^{x+2}\)
 \(f(x)=(2.3)^{x}3\)
 \(f(x)=e^{x}+5\)
 \(f(x)=e^{x}\)
 Answer

1.
3.
5.
7.
In the following exercises, solve each equation.
 \(3^{5 x6}=81\)
 \(2^{x^{2}}=16\)
 \(9^{x}=27\)
 \(5^{x^{2}+2 x}=\frac{1}{5}\)
 \(e^{4 x} \cdot e^{7}=e^{19}\)
 \(\frac{e^{x^{2}}}{e^{15}}=e^{2 x}\)
 Answer

2. \(x=2, x=2\)
4. \(x=1\)
6. \(x=3, x=5\)
In the following exercises, solve.
 Felix invested $\(12,000\) in a savings account. If the interest rate is \(4\)% how much will be in the account in \(12\) years by each method of compounding?
 compound quarterly
 compound monthly
 compound continuously
 Sayed deposits $\(20,000\) in an investment account. What will be the value of his investment in \(30\) years if the investment is earning \(7\)% per year and is compounded continuously?
 A researcher at the Center for Disease Control and Prevention is studying the growth of a bacteria. She starts her experiment with \(150\) of the bacteria that grows at a rate of \(15\)% per hour. She will check on the bacteria every \(24\) hours. How many bacteria will he find in \(24\) hours?
 In the last five years the population of the United States has grown at a rate of \(0.7\)% per year to about \(318,900,000\). If this rate continues, what will be the population in \(5\) more years?
 Answer

2. \(\$ 163,323.40\)
4. \(330,259,000\)
Evaluate and Graph Logarithmic Functions
In the following exercises, convert from exponential to logarithmic form.
 \(5^{4}=625\)
 \(10^{3}=\frac{1}{1,000}\)
 \(63^{\frac{1}{5}}=\sqrt[5]{63}\)
 \(e^{y}=16\)
 Answer

2. \(\log \frac{1}{1,000}=3\)
4. \(\ln 16=y\)
In the following exercises, convert each logarithmic equation to exponential form.
 \(7=\log _{2} 128\)
 \(5=\log 100,000\)
 \(4=\ln x\)
 Answer

2. \(100000=10^{5}\)
In the following exercises, solve for \(x\).
 \(\log _{x} 125=3\)
 \(\log _{7} x=2\)
 \(\log _{\frac{1}{2}} \frac{1}{16}=x\)
 Answer

1. \(x=5\)
3. \(x=4\)
In the following exercises, find the exact value of each logarithm without using a calculator.
 \(\log _{2} 32\)
 \(\log _{8} 1\)
 \(\log _{3} \frac{1}{9}\)
 Answer

2. \(0\)
In the following exercises, graph each logarithmic function.
 \(y=\log _{5} x\)
 \(y=\log _{\frac{1}{4}} x\)
 \(y=\log _{0.8} x\)
 Answer

1.
3.
In the following exercises, solve each logarithmic equation.
 \(\log _{a} 36=5\)
 \(\ln x=3\)
 \(\log _{2}(5 x7)=3\)
 \(\ln e^{3 x}=24\)
 \(\log \left(x^{2}21\right)=2\)
 Answer

2. \(x=e^{3}\)
4. \(x=8\)
What is the decibel level of a train whistle with intensity \(10^{−3}\) watts per square inch?
 Answer

\(90\) dB
Use the Properties of Logarithms
In the following exercises, use the properties of logarithms to evaluate.

 \(\log _{7} 1\)
 \(\log _{12} 12\)

 \(5^{\log _{5} 13}\)
 \(\log _{3} 3^{9}\)

 \(10^{\log \sqrt{5}}\)
 \(\log 10^{3}\)

 \(e^{\ln 8}\)
 \(\ln e^{5}\)
 Answer

2.
 \(13\)
 \(9\)
4.
 \(8\)
 \(5\)
In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
 \(\log _{4}(64 x y)\)
 \(\log 10,000 m\)
 Answer

2. \(4+\log m\)
In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.
 \(\log _{7} \frac{49}{y}\)
 \(\ln \frac{e^{5}}{2}\)
 Answer

2. \(5\ln 2\)
In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.
 \(\log x^{9}\)
 \(\log _{4} \sqrt[7]{z}\)
 Answer

2. \(\frac{1}{7} \log _{4} z\)
In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
 \(\log _{3}\left(\sqrt{4} x^{7} y^{8}\right)\)
 \(\log _{5} \frac{8 a^{2} b^{6} c}{d^{3}}\)
 \(\ln \frac{\sqrt{3 x^{2}y^{2}}}{z^{4}}\)
 \(\log _{6} \sqrt[3]{\frac{7 x^{2}}{6 y^{3} z^{5}}}\)
 Answer

2. \(\begin{array}{l}{\log _{5} 8+2 \log _{5} a+6 \log _{5} b} {+\log _{5} c3 \log _{5} d}\end{array}\)
4. \(\begin{array}{l}{\frac{1}{3}\left(\log _{6} 7+2 \log _{6} x13 \log _{6} y\right.} {5 \log _{6} z )}\end{array}\)
In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
 \(\log _{2} 56\log _{2} 7\)
 \(3 \log _{3} x+7 \log _{3} y\)
 \(\log _{5}\left(x^{2}16\right)2 \log _{5}(x+4)\)
 \(\frac{1}{4} \log y2 \log (y3)\)
 Answer

2. \(\log _{3} x^{3} y^{7}\)
4. \(\log \frac{\sqrt[4]{y}}{(y3)^{2}}\)
In the following exercises, rounding to three decimal places, approximate each logarithm.
 \(\log _{5} 97\)
 \(\log _{\sqrt{3}} 16\)
 Answer

2. \(5.047\)
Solve Exponential and Logarithmic Equations
In the following exercises, solve for \(x\).
 \(3 \log _{5} x=\log _{5} 216\)
 \(\log _{2} x+\log _{2}(x2)=3\)
 \(\log (x1)\log (3 x+5)=\log x\)
 \(\log _{4}(x2)+\log _{4}(x+5)=\log _{4} 8\)
 \(\ln (3 x2)=\ln (x+4)+\ln 2\)
 Answer

2. \(x=4\)
4. \(x=3\)
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.
 \(2^{x}=101\)
 \(e^{x}=23\)
 \(\left(\frac{1}{3}\right)^{x}=7\)
 \(7 e^{x+3}=28\)
 \(e^{x4}+8=23\)
 Answer

1. \(x=\frac{\log 101}{\log 2} \approx 6.658\)
3. \(x=\frac{\log 7}{\log \frac{1}{3}} \approx1.771\)
5. \(x=\ln 15+4 \approx 6.708\)
 Jerome invests $\(18,000\) at age \(17\). He hopes the investments will be worth $\(30,000\) when he turns \(26\). If the interest compounds continuously, approximately what rate of growth will he need to achieve his goal? Is that a reasonable expectation?
 Elise invests $\(4500\) in an account that compounds interest monthly and earns \(6\)%.How long will it take for her money to double?
 Researchers recorded that a certain bacteria population grew from \(100\) to \(300\) in \(8\) hours. At this rate of growth, how many bacteria will there be in \(24\) hours?
 Mouse populations can double in \(8\) months \(\left(A=2 A_{0}\right)\). How long will it take for a mouse population to triple?
 The halflife of radioactive iodine is \(60\) days. How much of a \(50\) mg sample will be left in \(40\) days?
 Answer

2. \(11.6\) years
4. \(12.7\) months
Practice Test
 For the functions, \(f(x)=6x+1\) and \(g(x)=8x−3\), find
 \((f \circ g)(x)\)
 \((g \circ f)(x)\)
 \((f \cdot g)(x)\)
 Determine if the following set of ordered pairs represents a function and if so, is the function onetoone. \(\{(2,2),(1,3),(0,1),(1,2),(2,3)\}\)
 Determine whether each graph is the graph of a function and if so, is it onetoone.
Figure 10.E.12
Figure 10.E.13
 Graph, on the same coordinate system, the inverse of the onetoone function shown.
5. Find the inverse of the function \(f(x)=x^{5}−9\).
6. Graph the function \(g(x)=2^{x3}\).
7. Solve the equation \(2^{2 x4}=64\).
8. Solve the equation \(\frac{e^{x^{2}}}{e^{4}}=e^{3 x}\).
9. Megan invested $\(21,000\) in a savings account. If the interest rate is \(5\)%, how much will be in the account in \(8\) years by each method of compounding?
 compound quarterly
 compound monthly
 compound continuously
10. Convert the equation from exponential to logarithmic form: \(10^{2}=\frac{1}{100}\).
11. Convert the equation from logarithmic equation to exponential form: \(3=\log _{7} 343\).
12. Solve for \(x\): \(\log _{5} x=3\)
13. Evaluate log \(_{11} 1\).
14. Evaluate \(\log _{4} \frac{1}{64}\).
15. Graph the function \(y=\log _{3} x\).
16. Solve for \(x\): \(\log \left(x^{2}39\right)=1\)
17. What is the decibel level of a small fan with intensity \(10^{−8}\) watts per square inch?
18. Evaluate each.
 \(6^{\log _{6} 17}\)
 \(\log _{9} 9^{3}\)
 Answer

1.
 \(48 x17\)
 \(48 x+5\)
 \(48 x^{2}10 x3\)
3.
 Not a function
 Onetoone function
5. \(f^{1}(x)=\sqrt[5]{x+9}\)
7. \(x=5\)
9.
 $\(31,250.74\)
 $\(31,302.29\)
 $\(31,328.32\)
11. \(343=7^{3}\)
13. \(0\)
15.
17. \(40\) dB
In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.
 \(\log _{5} 25 a b\)
 \(\ln \frac{e^{12}}{8}\)
 \(\log _{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}\)
 Answer

1. \(2+\log _{5} a+\log _{5} b\)
3. \(\begin{array}{l}{\frac{1}{4}\left(\log _{2} 5+3 \log _{2} x42 \log _{2} y\right.} {7 \log _{2} z )}\end{array}\)
In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible.
 \(5 \log _{4} x+3 \log _{4} y\)
 \(\frac{1}{6} \log x3 \log (x+5)\)
 Rounding to three decimal places, approximate \(\log _{4} 73\).
 Solve for \(x\): \(\log _{7}(x+2)+\log _{7}(x3)=\log _{7} 24\)
 Answer

2. \(\log \frac{\sqrt[6]{x}}{(x+5)^{3}}\)
4. \(x=6\)
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.
 \(\left(\frac{1}{5}\right)^{x}=9\)
 \(5 e^{x4}=40\)
 Jacob invests $\(14,000\) in an account that compounds interest quarterly and earns \(4\)%. How long will it take for his money to double?
 Researchers recorded that a certain bacteria population grew from \(500\) to \(700\) in \(5\) hours. At this rate of growth, how many bacteria will there be in \(20\) hours?
 A certain beetle population can double in \(3\) months \(\left(A=2 A_{0}\right)\). How long will it take for that beetle population to triple?
 Answer

2. \(x=\ln 8+4 \approx 6.079\)
4. \(1,921\) bacteria