Chapter 4.3 Practice

Questions

  1. Graph the following function. Identify the domain, range, and asymptotes of each function.
    1. \(f(x)=-4^{x+2}+1\)
    2. \(f(x)=3^{x}-5\)
    3. \(f(x)=\left(\frac{1}{2}\right)^{3x}\)
    4. \(f(x)=2e^{x+3}-1\)
  2. Take \(f(x)=-4^{x+2}+1\), \(g(x)=3^{x}-5\), \(h(x)=\left(\frac{1}{2}\right)^{3x},\) and \(j(x)=2e^{x+3}-1.\) Solve the following inequalities.
    1. \(f(x)>1\)
    2. \(g(x)>-5\)
    3. \(h(x)\leq 0\)
    4. \(j(x)\geq -1\)
  3. Compute the following values.
    1. \(\log_2(4)\)
    2. \(\log_3(27)\)
    3. \(\log_4(16)\)
    4. \(\log_\frac{1}{3}(9)\)
    5. \(\log_\frac{1}{3}\left(\frac{1}{3}\right)\)
    6. \(\log_4(1)\)
    7. \(\log_\frac{1}{2}(4)\)
    8. \(\log_e\left(\frac{1}{e^2}\right)\)
    9. \(\ln\left(e^3\right)\)
  4. Graph the following function. Identify the domain, range, and asymptotes of each function.
    1. \(f(x)=\ln(-x+3)-1\)
    2. \(f(x)=3\log_2(x+1)-5\)
    3. \(f(x)=\log_\frac{1}{4}(-x-5)\)
  5. Solve the following equations.
    1. \(4^{x}-16=0\)
    2. \(3^{x}-11=0\)
    3. \(6^{x-1}-36^{x}=0\)
    4. \(e^{2x}-\frac{1}{e^{x-2}}=0\)
    5. \(\log_2(x)+\log_2(x-4)=0\)
    6. \(\ln(-x)-\ln(-x-1)=1\)
  6. Take \(a\) and \(b\) to be two real numbers so that \(\log_2(a)=15\) and \(\log_2(b)=-\frac{3}{2}.\) Compute the following.
    1. \(\log_2(ab)\)
    2. \(\log_2(16a^2b)\)
    3. \(\log_2(a^5b^3)\)
    4. \(\log_2(\frac{4a^3}{b^7})\)
    5. \(\log_2(\frac{a}{16b^2})\)
  7. A quantity \(A\) changes according to a linear model for change and \[\begin{cases} A(0)=3\\ A(1)=13. \end{cases}\] Identify a formula for \(A(t).\)
  8. A quantity \(A\) changes according to an exponential model for change and \[\begin{cases} A(0)=11\\ A(1)=12. \end{cases}\] Identify a formula for \(A(t).\)
  9. A quantity \(A\) changes according to a linear model for change and \[\begin{cases} A(3)=4\\ A(19)=12. \end{cases}\] Identify a formula for \(A(t).\)
  10. A quantity \(A\) changes according to an exponential model for change and \[\begin{cases} A(1)=5\\ A(9)=11. \end{cases}\] Identify a formula for \(A(t).\)
  11. At time \(2\) an experimenter has \(15\) grams of a radioactive substance. At time \(12,\) \(2\) grams remain.
    1. Determine the half-life of the substance.
    2. Determine the time that it takes for only a fraction of \(\frac{1}{5}\) of the original remains.
    3. Determine the decay rate of the substance.
  12. A mass of bacteria experiences exponential growth. At time \(4\) an experimenter has \(100\) grams of bacteria. At time \(12\), the mass has grown to a mass of \(200\).
    1. Determine the doubling time of the bacteria.
    2. Determine the time that it takes for the amount of the material to increase by a factor of \(3\).
    3. Determine the growth rate of the substance.

Answers

    1. Domain is \((-\infty,\infty)\), range is \((-\infty,-1)\) and horizontal asymptote is \(y=1\).
    2. Domain is \((-\infty,\infty)\), range is \((-5,\infty)\) and horizontal asymptote is \(y=-5\).
    3. Domain is \((-\infty,\infty)\), range is \((0,\infty)\) and horizontal asymptote is \(y=0\).
    4. Domain is \((-\infty,\infty)\), range is \((0,\infty)\) and horizontal asymptote is \(y=0\).
    1. No solution
    2. \((-\infty,\infty)\)
    3. No solution
    4. \((-\infty,\infty)\)
    1. \(2\)
    2. \(3\)
    3. \(2\)
    4. \(-2\)
    5. \(1\)
    6. \(0\)
    7. \(-2\)
    8. \(-2\)
    9. \(3\)
    1. Domain is \((-\infty,3)\), range is \((-\infty,\infty)\) and vertical asymptote is \(x=3\).
    2. Domain is \((-1,\infty)\), range is \((-\infty,\infty)\) and vertical asymptote is \(x=-1\).
    3. Domain is \((-\infty,-5)\), range is \((-\infty,\infty)\) and vertical asymptote is \(x=-5\).
    1. \(2\)
    2. \(\log_3(11)\)
    3. \(-1\)
    4. \(\frac{2}{3}\)
    5. \(2+\sqrt{5}\)
    6. \(-\frac{e}{e-1}\)
    1. \(\frac{27}{3}\)
    2. \(\frac{65}{2}\)
    3. \(\frac{141}{2}\)
    4. \(\frac{115}{2}\)
    5. \(14\)
  7. \(A(t)=10t+3\)
  8. \(A(t)=11\left(\frac{12}{11}\right)^t\)
  9. \(A(t)=8\frac{(t-3)}{16}+4\)
  10. \(A(t)=5\left(\frac{11}{5}\right)^{\frac{t-1}{8}}\)
    1. \(\frac{10\ln\left(\frac{1}{2}\right)}{\ln\left(\frac{2}{15}\right)}\)
    2. \(\frac{10\ln\left(\frac{1}{5}\right)}{\ln\left(\frac{2}{15}\right)}\)
    3. \(-\frac{1}{10}\ln\left(\frac{2}{15}\right)\)
    1. \(8\)
    2. \(\frac{8\ln\left(3\right)}{\ln\left(2\right)}\)
    3. \(\frac{1}{8}\ln(2)\)

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