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Mathematics LibreTexts

Chapter 10 Review Exercises

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    Chapter Review Exercises

    Finding Composite and Inverse Functions

    Exercise \(\PageIndex{1}\) Find and Evaluate Composite Functions

    In the following exercises, for each pair of functions, find

    1. \((f \circ g)(x)\)
    2. \((g \circ f)(x)\)
    3. \((f \cdot g)(x)\)

    1. \(f(x)=7 x-2\) and \(g(x)=5 x+1\)

    2. \(f(x)=4 x\) and \(g(x)=x^{2}+3 x\)

    Answer

    2.

    1. \(4 x^{2}+12 x\)
    2. \(16 x^{2}+12 x\)
    3. \(4 x^{3}+12 x^{2}\)
    Exercise \(\PageIndex{2}\) Find and Evaluate Composite Functions

    In the following exercises, evaluate the composition.

    1. For functions \(f(x)=3 x^{2}+2\) and \(g(x)=4 x-3\), find
      1. \((f \circ g)(-3)\)
      2. \((g \circ f)(-2)\)
      3. \((f \circ f)(-1)\)
    2. For functions \(f(x)=2 x^{3}+5\) and \(g(x)=3 x^{2}-7\), find
      1. \((f \circ g)(-1)\)
      2. \((g \circ f)(-2)\)
      3. \((g \circ g)(1)\)
    Answer

    2.

    1. \(-123\)
    2. \(356\)
    3. \(41\)
    Exercise \(\PageIndex{3}\) Determine Whether a Function is One-to-One

    In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.

    1. \(\begin{array}{l}{\{(-3,-5),(-2,-4),(-1,-3),(0,-2)} , {(-1,-1),(-2,0),(-3,1) \}}\end{array}\)
    2. \(\begin{array}{l}{\{(-3,0),(-2,-2),(-1,0),(0,1)} , {(1,2),(2,1),(3,-1) \}}\end{array}\)
    3. \(\begin{array}{l}{\{(-3,3),(-2,1),(-1,-1),(0,-3)} , {(1,-5),(2,-4),(3,-2) \}}\end{array}\)
    Answer

    2. Function; not one-to-one

    Exercise \(\PageIndex{4}\) Determine Whether a Function is One-to-One

    In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.


      1. This figure shows a line from (negative 6, negative 2) up to (negative 1, 3) and then down from there to (6, negative 4).
        Figure 10.E.1

      2. This figure shows a line from (6, 5) down to (0, negative 1) and then down from there to (5, negative 6).
        Figure 10.E.2

      1. This figure shows a curved line from (negative 6, negative 2) up to the origin and then continuing up from there to (6, 2).
        Figure 10.E.3

      2. This figure shows a circle of radius 2 with center at the origin.
        Figure 10.E.4
    Answer

    1.

    1. Function; not one-to-one
    2. Not a function
    Exercise \(\PageIndex{5}\) Find the Inverse of a Function

    In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.

    1. \(\{(-3,10),(-2,5),(-1,2),(0,1)\}\)
    Answer

    1. Inverse function: \(\{(10,-3),(5,-2),(2,-1),(1,0)\}\). Domain: \(\{1,2,5,10\}\). Range: \(\{-3,-2,-1,0\}\).

    Exercise \(\PageIndex{6}\) Find the Inverse of a Function

    In the following exercise, graph the inverse of the one-to-one function shown.

    This figure shows a line segment from (negative 4, negative 2) up to (negative 2, 1) then up to (2, 2) and then up to (3, 4).
    Figure 10.E.5
    Answer

    Solve on your own

    Exercise \(\PageIndex{7}\) Find the Inverse of a Function

    In the following exercises, verify that the functions are inverse functions.

    1. \(\begin{array}{l}{f(x)=3 x+7 \text { and }} {g(x)=\frac{x-7}{3}}\end{array}\)
    2. \(\begin{array}{l}{f(x)=2 x+9 \text { and }} {g(x)=\frac{x+9}{2}}\end{array}\)
    Answer

    1. \(g(f(x))=x,\) and \(f(g(x))=x,\) so they are inverses.

    Exercise \(\PageIndex{8}\) Find the Inverse of a Function
    1. \(f(x)=6 x-11\)
    2. \(f(x)=x^{3}+13\)
    3. \(f(x)=\frac{1}{x+5}\)
    4. \(f(x)=\sqrt[5]{x-1}\)
    Answer

    1. \(f^{-1}(x)=\frac{x+11}{6}\)

    3. \(f^{-1}(x)=\frac{1}{x}-5\)

    Evaluate and Graph Exponential Functions

    Exercise \(\PageIndex{9}\) Graph Exponential Functions

    In the following exercises, graph each of the following functions.

    1. \(f(x)=4^{x}\)
    2. \(f(x)=\left(\frac{1}{5}\right)^{x}\)
    3. \(g(x)=(0.75)^{x}\)
    4. \(g(x)=3^{x+2}\)
    5. \(f(x)=(2.3)^{x}-3\)
    6. \(f(x)=e^{x}+5\)
    7. \(f(x)=-e^{x}\)
    Answer

    1.

    This figure shows an exponential line passing through the points (negative 1, 1 over 4), (0, 1), and (1, 4).
    Figure 10.E.6

    3.

    This figure shows an exponential line passing through the points (negative 1, 4 over 3), (0, 1), and (1, 3 over 4).
    Figure 10.E.7

    5.

    This figure shows an exponential line passing through the points (negative 1, negative 59 over 23), (0, negative 2), and (1, negative7 over 10).
    Figure 10.E.8

    7.

    This figure shows an exponential line passing through the points (negative 1, negative 1 over e), (0, negative 1), and (1, negative e).
    Figure 10.E.9
    Exercise \(\PageIndex{10}\) Solve Exponential Equations

    In the following exercises, solve each equation.

    1. \(3^{5 x-6}=81\)
    2. \(2^{x^{2}}=16\)
    3. \(9^{x}=27\)
    4. \(5^{x^{2}+2 x}=\frac{1}{5}\)
    5. \(e^{4 x} \cdot e^{7}=e^{19}\)
    6. \(\frac{e^{x^{2}}}{e^{15}}=e^{2 x}\)
    Answer

    2. \(x=-2, x=2\)

    4. \(x=-1\)

    6. \(x=-3, x=5\)

    Exercise \(\PageIndex{11}\) Use Exponential Models in Applications

    In the following exercises, solve.

    1. 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?
      1. compound quarterly
      2. compound monthly
      3. compound continuously
    2. 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?
    3. 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?
    4. 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

    Exercise \(\PageIndex{12}\) Convert Between Exponential and Logarithmic Form

    In the following exercises, convert from exponential to logarithmic form.

    1. \(5^{4}=625\)
    2. \(10^{-3}=\frac{1}{1,000}\)
    3. \(63^{\frac{1}{5}}=\sqrt[5]{63}\)
    4. \(e^{y}=16\)
    Answer

    2. \(\log \frac{1}{1,000}=-3\)

    4. \(\ln 16=y\)

    Exercise \(\PageIndex{13}\) Convert Between Exponential and Logarithmic Form

    In the following exercises, convert each logarithmic equation to exponential form.

    1. \(7=\log _{2} 128\)
    2. \(5=\log 100,000\)
    3. \(4=\ln x\)
    Answer

    2. \(100000=10^{5}\)

    Exercise \(\PageIndex{14}\) Evaluate Logarithmic Functions

    In the following exercises, solve for \(x\).

    1. \(\log _{x} 125=3\)
    2. \(\log _{7} x=-2\)
    3. \(\log _{\frac{1}{2}} \frac{1}{16}=x\)
    Answer

    1. \(x=5\)

    3. \(x=4\)

    Exercise \(\PageIndex{15}\) Evaluate Logarithmic Functions

    In the following exercises, find the exact value of each logarithm without using a calculator.

    1. \(\log _{2} 32\)
    2. \(\log _{8} 1\)
    3. \(\log _{3} \frac{1}{9}\)
    Answer

    2. \(0\)

    Exercise \(\PageIndex{16}\) Graph Logarithmic Functions

    In the following exercises, graph each logarithmic function.

    1. \(y=\log _{5} x\)
    2. \(y=\log _{\frac{1}{4}} x\)
    3. \(y=\log _{0.8} x\)
    Answer

    1.

    This figure shows a logarithmic line passing through the points (1 over 5, negative 1), (1, 0), and (5, 1).
    Figure 10.E.10

    3.

    This figure shows a logarithmic line passing through the points (4 over 5, 1), (1, 0), and (5 over 4, negative 1).
    Figure 10.E.11
    Exercise \(\PageIndex{17}\) Solve Logarithmic Equations

    In the following exercises, solve each logarithmic equation.

    1. \(\log _{a} 36=5\)
    2. \(\ln x=-3\)
    3. \(\log _{2}(5 x-7)=3\)
    4. \(\ln e^{3 x}=24\)
    5. \(\log \left(x^{2}-21\right)=2\)
    Answer

    2. \(x=e^{-3}\)

    4. \(x=8\)

    Exercise \(\PageIndex{18}\) Use Logarithmic Models in Applications

    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

    Exercise \(\PageIndex{19}\) Use the Properties of Logarithms

    In the following exercises, use the properties of logarithms to evaluate.

      1. \(\log _{7} 1\)
      2. \(\log _{12} 12\)
      1. \(5^{\log _{5} 13}\)
      2. \(\log _{3} 3^{-9}\)
      1. \(10^{\log \sqrt{5}}\)
      2. \(\log 10^{-3}\)
      1. \(e^{\ln 8}\)
      2. \(\ln e^{5}\)
    Answer

    2.

    1. \(13\)
    2. \(-9\)

    4.

    1. \(8\)
    2. \(5\)
    Exercise \(\PageIndex{20}\) Use the Properties of Logarithms

    In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

    1. \(\log _{4}(64 x y)\)
    2. \(\log 10,000 m\)
    Answer

    2. \(4+\log m\)

    Exercise \(\PageIndex{21}\) Use the Properties of Logarithms

    In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.

    1. \(\log _{7} \frac{49}{y}\)
    2. \(\ln \frac{e^{5}}{2}\)
    Answer

    2. \(5-\ln 2\)

    Exercise \(\PageIndex{22}\) Use the Properties of Logarithms

    In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.

    1. \(\log x^{-9}\)
    2. \(\log _{4} \sqrt[7]{z}\)
    Answer

    2. \(\frac{1}{7} \log _{4} z\)

    Exercise \(\PageIndex{23}\) Use the Properties of Logarithms

    In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.

    1. \(\log _{3}\left(\sqrt{4} x^{7} y^{8}\right)\)
    2. \(\log _{5} \frac{8 a^{2} b^{6} c}{d^{3}}\)
    3. \(\ln \frac{\sqrt{3 x^{2}-y^{2}}}{z^{4}}\)
    4. \(\log _{6} \sqrt[3]{\frac{7 x^{2}}{6 y^{3} z^{5}}}\)
    Answer

    2. \(\begin{array}{l}{\log _{5} 8+2 \log _{5} a+6 \log _{5} b} {+\log _{5} c-3 \log _{5} d}\end{array}\)

    4. \(\begin{array}{l}{\frac{1}{3}\left(\log _{6} 7+2 \log _{6} x-1-3 \log _{6} y\right.} {-5 \log _{6} z )}\end{array}\)

    Exercise \(\PageIndex{24}\) Use the Properties of Logarithms

    In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.

    1. \(\log _{2} 56-\log _{2} 7\)
    2. \(3 \log _{3} x+7 \log _{3} y\)
    3. \(\log _{5}\left(x^{2}-16\right)-2 \log _{5}(x+4)\)
    4. \(\frac{1}{4} \log y-2 \log (y-3)\)
    Answer

    2. \(\log _{3} x^{3} y^{7}\)

    4. \(\log \frac{\sqrt[4]{y}}{(y-3)^{2}}\)

    Exercise \(\PageIndex{25}\) Use the Change-of-Base Formula

    In the following exercises, rounding to three decimal places, approximate each logarithm.

    1. \(\log _{5} 97\)
    2. \(\log _{\sqrt{3}} 16\)
    Answer

    2. \(5.047\)

    Solve Exponential and Logarithmic Equations

    Exercise \(\PageIndex{26}\) Solve Logarithmic Equations Using the Properties of Logarithms

    In the following exercises, solve for \(x\).

    1. \(3 \log _{5} x=\log _{5} 216\)
    2. \(\log _{2} x+\log _{2}(x-2)=3\)
    3. \(\log (x-1)-\log (3 x+5)=-\log x\)
    4. \(\log _{4}(x-2)+\log _{4}(x+5)=\log _{4} 8\)
    5. \(\ln (3 x-2)=\ln (x+4)+\ln 2\)
    Answer

    2. \(x=4\)

    4. \(x=3\)

    Exercise \(\PageIndex{27}\) Solve Exponential Equations Using Logarithms

    In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

    1. \(2^{x}=101\)
    2. \(e^{x}=23\)
    3. \(\left(\frac{1}{3}\right)^{x}=7\)
    4. \(7 e^{x+3}=28\)
    5. \(e^{x-4}+8=23\)
    Answer

    1. \(x=\frac{\log 101}{\log 2} \approx 6.658\)

    3. \(x=\frac{\log 7}{\log \frac{1}{3}} \approx-1.771\)

    5. \(x=\ln 15+4 \approx 6.708\)

    Exercise \(\PageIndex{28}\) Use Exponential Models in Applications
    1. 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?
    2. Elise invests $\(4500\) in an account that compounds interest monthly and earns \(6\)%.How long will it take for her money to double?
    3. 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?
    4. Mouse populations can double in \(8\) months \(\left(A=2 A_{0}\right)\). How long will it take for a mouse population to triple?
    5. The half-life 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

    Exercise \(\PageIndex{29}\)
    1. For the functions, \(f(x)=6x+1\) and \(g(x)=8x−3\), find
      1. \((f \circ g)(x)\)
      2. \((g \circ f)(x)\)
      3. \((f \cdot g)(x)\)
    2. Determine if the following set of ordered pairs represents a function and if so, is the function one-to-one. \(\{(-2,2),(-1,-3),(0,1),(1,-2),(2,-3)\}\)
    3. Determine whether each graph is the graph of a function and if so, is it one-to-one.

      1. This figure shows a parabola opening to the right with vertex (negative 3, 0).
        Figure 10.E.12

      2. This figure shows an exponential line passing through the points (negative 1, 1 over 2), (0, 1), and (1, 2).
        Figure 10.E.13
    4. Graph, on the same coordinate system, the inverse of the one-to-one function shown.
    This figure shows a line segment passing from the point (negative 3, 3) to (negative 1, 2) to (0, negative 2) to (2, negative 4).
    Figure 10.E.14

    5. Find the inverse of the function \(f(x)=x^{5}−9\).

    6. Graph the function \(g(x)=2^{x-3}\).

    7. Solve the equation \(2^{2 x-4}=64\).

    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?

    1. compound quarterly
    2. compound monthly
    3. 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.

    1. \(6^{\log _{6} 17}\)
    2. \(\log _{9} 9^{-3}\)
    Answer

    1.

    1. \(48 x-17\)
    2. \(48 x+5\)
    3. \(48 x^{2}-10 x-3\)

    3.

    1. Not a function
    2. One-to-one function

    5. \(f^{-1}(x)=\sqrt[5]{x+9}\)

    7. \(x=5\)

    9.

    1. $\(31,250.74\)
    2. $\(31,302.29\)
    3. $\(31,328.32\)

    11. \(343=7^{3}\)

    13. \(0\)

    15.

    This figure shows a logarithmic line passing through (1 over 3, 1), (1, 0), and (3, 1).
    Figure 10.E.15

    17. \(40\) dB

    Exercise \(\PageIndex{30}\)

    In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.

    1. \(\log _{5} 25 a b\)
    2. \(\ln \frac{e^{12}}{8}\)
    3. \(\log _{2} \sqrt[4]{\frac{5 x^{3}}{16 y^{2} z^{7}}}\)
    Answer

    1. \(2+\log _{5} a+\log _{5} b\)

    3. \(\begin{array}{l}{\frac{1}{4}\left(\log _{2} 5+3 \log _{2} x-4-2 \log _{2} y\right.} {-7 \log _{2} z )}\end{array}\)

    Exercise \(\PageIndex{31}\)

    In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible.

    1. \(5 \log _{4} x+3 \log _{4} y\)
    2. \(\frac{1}{6} \log x-3 \log (x+5)\)
    3. Rounding to three decimal places, approximate \(\log _{4} 73\).
    4. Solve for \(x\): \(\log _{7}(x+2)+\log _{7}(x-3)=\log _{7} 24\)
    Answer

    2. \(\log \frac{\sqrt[6]{x}}{(x+5)^{3}}\)

    4. \(x=6\)

    Exercise \(\PageIndex{32}\)

    In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.

    1. \(\left(\frac{1}{5}\right)^{x}=9\)
    2. \(5 e^{x-4}=40\)
    3. Jacob invests $\(14,000\) in an account that compounds interest quarterly and earns \(4\)%. How long will it take for his money to double?
    4. 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?
    5. 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


    Chapter 10 Review Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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