Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

10.3E: Exercises

  • Page ID
    30569
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

    Practice Makes Perfect

    Exercise \(\PageIndex{17}\) Graph Exponential Functions

    In the following exercises, graph each exponential function.

    1. \(f(x)=2^{x}\)
    2. \(g(x)=3^{x}\)
    3. \(f(x)=6^{x}\)
    4. \(g(x)=7^{x}\)
    5. \(f(x)=(1.5)^{x}\)
    6. \(g(x)=(2.5)^{x}\)
    7. \(f(x)=\left(\frac{1}{2}\right)^{x}\)
    8. \(g(x)=\left(\frac{1}{3}\right)^{x}\)
    9. \(f(x)=\left(\frac{1}{6}\right)^{x}\)
    10. \(g(x)=\left(\frac{1}{7}\right)^{x}\)
    11. \(f(x)=(0.4)^{x}\)
    12. \(g(x)=(0.6)^{x}\)
    Answer

    1.

    This figure shows a curve that passes through (negative 1, 1 over 2) through (0, 1) to (1, 2).
    Figure 10.2.22

    3.

    This figure shows a curve that passes through (negative 1, 1 over 6) through (0, 1) to (1, 6).
    Figure 10.2.23

    5.

    This figure shows a curve that passes through (negative 1, 2 over 3) through (0, 1) to (1, 3 over 2).
    Figure 10.2.24

    7.

    This figure shows a curve that passes through (negative 1, 2) through (0, 1) to (1, 1 over 2).
    Figure 10.2.25

    9.

    This figure shows a curve that passes through (negative1, 6) through (0, 1) to (1, 1 over 6).
    Figure 10.2.26

    11.

    This figure shows a curve that passes through (negative 1, 5 over 2) through (0, 1) to (1, 2 over 5).
    Figure 10.2.27
    Exercise \(\PageIndex{18}\) Graph Exponential Functions

    In the following exercises, graph each function in the same coordinate system.

    1. \(f(x)=4^{x}, g(x)=4^{x-1}\)
    2. \(f(x)=3^{x}, g(x)=3^{x-1}\)
    3. \(f(x)=2^{x}, g(x)=2^{x-2}\)
    4. \(f(x)=2^{x}, g(x)=2^{x+2}\)
    5. \(f(x)=3^{x}, g(x)=3^{x}+2\)
    6. \(f(x)=4^{x}, g(x)=4^{x}+2\)
    7. \(f(x)=2^{x}, g(x)=2^{x}+1\)
    8. \(f(x)=2^{x}, g(x)=2^{x}-1\)
    Answer

    1.

    This figure shows two functions. The first function f of x equals 4 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 4), (0, 1) and (1, 4). The second function g of x equals 4 to the x minus 1 power is marked in red and passes through the points (0, 1 over 4), (1, 1) and (2, 4).
    Figure 10.2.28

    3.

    This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second function g of x equals 2 to the x minus 2 power is marked in red and passes through the points (0, 1 over 4), (1, 1 over 2), and (2, 1).
    Figure 10.2.29

    5.

    This figure shows two functions. The first function f of x equals 3 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 3), (0, 1), and (1, 3). The second function g of x equals 3 to the x power plus 2 is marked in red and passes through the points (negative 2, 1), (negative 1, 3), and (0, 5).
    Figure 10.2.30

    7.

    This figure shows two functions. The first function f of x equals 2 to the x power is marked in blue and corresponds to a curve that passes through the points (negative 1, 1 over 2), (0, 1), and (1, 2). The second function g of x equals 2 to the x power plus 1 is marked in red and passes through the points (negative 1, 1), (0, 2), and (1, 4).
    Figure 10.2.31
    Exercise \(\PageIndex{19}\) Graph Exponential Functions

    In the following exercises, graph each exponential function.

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

    1.

    This figure shows an exponential curve that passes through (negative 3, 1 over 3), (negative 2, 1), and (0, 9).
    Figure 10.2.32

    3.

    This figure shows an exponential that passes through (negative 1, 7 over 2), (0, 4), and (1, 5).
    Figure 10.2.33

    5.

    This figure shows an exponential that passes through (2, 4), (3, 2), and (4, 1).
    Figure 10.2.34

    7.

    This figure shows an exponential that passes through (1, 1 plus 1 over e), (0, 2), and (1, e).
    Figure 10.2.35

    9.

    This figure shows an exponential that passes through (negative 1, negative 1 over 2), (0, negative 1), and (1, 2).
    Figure 10.2.36
    Exercise \(\PageIndex{20}\) Solve Exponential Equations

    In the following exercises, solve each equation.

    1. \(2^{3 x-8}=16\)
    2. \(2^{2 x-3}=32\)
    3. \(3^{x+3}=9\)
    4. \(3^{x^{2}}=81\)
    5. \(4^{x^{2}}=4\)
    6. \(4^{x}=32\)
    7. \(4^{x+2}=64\)
    8. \(4^{x+3}=16\)
    9. \(2^{x^{2}+2 x}=\frac{1}{2}\)
    10. \(3^{x^{2}-2 x}=\frac{1}{3}\)
    11. \(e^{3 x} \cdot e^{4}=e^{10}\)
    12. \(e^{2 x} \cdot e^{3}=e^{9}\)
    13. \(\frac{e^{x^{2}}}{e^{2}}=e^{x}\)
    14. \(\frac{e^{x^{2}}}{e^{3}}=e^{2 x}\)
    Answer

    1. \(x=4\)

    3. \(x=-1\)

    5. \(x=-1, x=1\)

    7. \(x=1\)

    9. \(x=-1\)

    11. \(x=2\)

    13. \(x=-1, x=2\)

    Exercise \(\PageIndex{21}\) Solve Exponential Equations

    In the following exercises, match the graphs to one of the following functions:

    1. \(2^{x}\)
    2. \(2^{x+1}\)
    3. \(2^{x-1}\)
    4. \(2^{x}+2\)
    5. \(2^{x}-2\)
    6. \(3^{x}\)

    1. This figure shows an exponential that passes through (1, 1 over 3), (0, 1), and (1, 3).
      Figure 10.2.37

    2. This figure shows an exponential that passes through (negative 2, 1 over 2), (negative 1, 1), and (0, 2).
      Figure 10.2.38

    3. This figure shows an exponential that passes through (1, 1 over 2), (0, 1), and (1, 2).
      Figure 10.2.39

    4. This figure shows an exponential that passes through (0, 1 over 2), (1, 1), and (2, 2).
      Figure 10.2.40

    5. This figure shows an exponential that passes through (negative 1, 3 over 2), (0, negative 1), and (1, 0).
      Figure 10.2.41

    6. This figure shows an exponential that passes through (negative 1, 5 over 2), (0, 3), and (1, 4).
      Figure 10.2.42
    Answer

    1. f

    3. a

    5. e

    Exercise \(\PageIndex{22}\) Use Exponential Models in Applications

    In the following exercises, use an exponential model to solve.

    1. Edgar accumulated $\(5,000\) in credit card debt. If the interest rate is \(20\)% per year, and he does not make any payments for \(2\) years, how much will he owe on this debt in \(2\) years by each method of compounding?
      1. compound quarterly
      2. compound monthly
      3. compound continuously
    2. Cynthia invested $\(12,000\) in a savings account. If the interest rate is \(6\)%, how much will be in the account in \(10\) years by each method of compounding?
      1. compound quarterly
      2. compound monthly
      3. compound continuously
    3. Rochelle deposits $\(5,000\) in an IRA. What will be the value of her investment in \(25\) years if the investment is earning \(8\)% per year and is compounded continuously?
    4. Nazerhy deposits $\(8,000\) in a certificate of deposit. The annual interest rate is \(6\)% and the interest will be compounded quarterly. How much will the certificate be worth in \(10\) years?
    5. A researcher at the Center for Disease Control and Prevention is studying the growth of a bacteria. He starts his experiment with \(100\) of the bacteria that grows at a rate of \(6\)% per hour. He will check on the bacteria every \(8\) hours. How many bacteria will he find in \(8\) hours?
    6. A biologist is observing the growth pattern of a virus. She starts with \(50\) of the virus that grows at a rate of \(20\)% per hour. She will check on the virus in \(24\) hours. How many viruses will she find?
    7. In the last ten years the population of Indonesia has grown at a rate of \(1.12\)% per year to \(258,316,051\). If this rate continues, what will be the population in \(10\) more years?
    8. In the last ten years the population of Brazil has grown at a rate of \(0.9\)% per year to \(205,823,665\). If this rate continues, what will be the population in \(10\) more years?
    Answer

    1.

    1. $\(7,387.28\)
    2. $\(7,434.57\)
    3. $\(7,459.12\)

    3. $\(36,945.28\)

    5. \(223\) bacteria

    7. \(288,929,825\)

    Exercise \(\PageIndex{23}\) Writing Exercises
    1. Explain how you can distinguish between exponential functions and polynomial functions.
    2. Compare and contrast the graphs of \(y=x^{2}\) and \(y=2^{x}\).
    3. What happens to an exponential function as the values of \(x\) decreases? Will the graph ever cross the \(x\)-axis? Explain.
    Answer

    1. Answers will vary

    3. Answers will vary

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    This table has four rows and four columns. The first row, which serves as a header, reads I can…, Confidently, With some help, and No—I don’t get it. The first column below the header row reads Graph exponential functions, solve exponential equations, and use exponential models in applications.
    Figure 10.2.43

    b. After reviewing this checklist, what will you do to become confident for all objectives?


    10.3E: 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.