Skip to main content

Precalculus Review Materials

Jane Butterfield, Gary MacGillivray

Section 4.6 Exercises

These are the exercises in Module 4.
🔗
Note that many of these exercises contain images, which might not display automatically. Make sure to click ’Activate’ on each in order to load the images.
🔗

Checkpoint 4.6.1.

Given that \(f(x)={5}x^{2} + {6}\) and \(g(x)= {4}x+{8}\) are functions from \(\mathbb{R}\) to \(\mathbb{R}\text{,}\) find
🔗
(a) \(f \circ g\text{.}\)
🔗
(b) \(g \circ f\text{.}\)
🔗
Answer 1.
\(80x^{2}+320x+326\)
🔗
🔗
Answer 2.
\(20x^{2}+24+8\)
🔗
🔗
🔗

Checkpoint 4.6.2.

Given that \(f(x)=7 x - 5\) and \(g(x)=7-x^2\text{,}\) calculate
🔗
(a) \((f\circ g)(-2)=\)
🔗
(b) \((g\circ f)(-2)=\)
🔗
Answer 1.
\(16\)
🔗
🔗
Answer 2.
\(-354\)
🔗
🔗
🔗

Checkpoint 4.6.3.

Given that \(f(x)=\frac{1}{x}\) and \(g(x)=8x + 8\text{,}\) calculate
🔗
(a) \((f\circ g)(x)=\), it’s domain is all real numbers except .
🔗
(b) \((g\circ f)(x)=\), it’s domain is all real numbers except .
🔗
(c) \((f\circ f)(x)=\), it’s domain is all real numbers except .
🔗
(d) \((g\circ g)(x)=\), it’s domain is \((\),\()\text{.}\)
🔗
Note: If needed enter \(\infty\) as infinity and \(-\infty\) as -infinity.
🔗
Answer 1.
\(\frac{1}{8x+8}\)
🔗
🔗
Answer 2.
\(-1\)
🔗
🔗
Answer 3.
\(\frac{8}{x}+8\)
🔗
🔗
Answer 4.
\(0\)
🔗
🔗
Answer 5.
\(x\)
🔗
🔗
Answer 6.
\(0\)
🔗
🔗
Answer 7.
\(8\mathopen{}\left(8x+8\right)+8\)
🔗
🔗
Answer 8.
\(-\infty \)
🔗
🔗
Answer 9.
\(\infty \)
🔗
🔗
🔗

Checkpoint 4.6.4.

Let \(\displaystyle f(x) = \frac{1}{x-7}\) and \(\displaystyle g(x) = \frac{5}{x}+7\text{.}\)
🔗
Then \((f\circ g)(x) =\) ,
🔗
\((g \circ f)(x ) =\) .
🔗
Answer 1.
\(\frac{x}{5}\)
🔗
🔗
Answer 2.
\(5x-35+7\)
🔗
🔗
🔗

Checkpoint 4.6.5.

Use properties of the functions to match each of the following functions with its graph. Do not use your calculator.
🔗
1. \({\frac{-1}{x^{2}}}\)
🔗
2. \({\frac{1}{x^{2}}}\)
🔗
3. \({\frac{1}{x}}\)
🔗
4. \({\frac{-1}{x}}\)
🔗
Answer 1.
\(\text{A}\)
🔗
🔗
Answer 2.
\(\text{B}\)
🔗
🔗
Answer 3.
\(\text{D}\)
🔗
🔗
Answer 4.
\(\text{C}\)
🔗
🔗
🔗

Checkpoint 4.6.6.

Determine the system of linear inequalities for which the solution set is shown on the graph.
🔗
  • \(\displaystyle \begin{cases} x \le 0 \\ x \le -2/y \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} x \ge 0 \\ x \ge -2/y \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} x \ge 0 \\ x \le -2/y \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} y \le -2/x \\ x \le 0 \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} y \le -5/x \\ x \ge 0 \end{cases} \)
    🔗
    🔗
  • None of the above
    🔗
    🔗
🔗
Answer.
\(\text{Choice 3}\)
🔗
🔗
🔗

Checkpoint 4.6.7.

Which of the regions A, B, C, D belong to the solution set of the given system of inequalities.
🔗
  • Region A lies above the curve and to the right of the \(y\)-axis.
    🔗
    🔗
  • Region B lies below the curve and to the right of the \(y\)-axis.
    🔗
    🔗
  • Region C lies below the curve and to the left of the \(y\)-axis.
    🔗
    🔗
  • Region D lies above the curve and to the left of the \(y\)-axis.
    🔗
    🔗
🔗
\(\begin{aligned} 2/x^2 \amp \le y \\ x \amp \ge 0 \end{aligned}\)
🔗
Answer.
\(\text{A}\)
🔗
🔗
🔗

Checkpoint 4.6.8.

Which of the regions A, B, C, D belong to the solution set of the given system of inequalities.
🔗
  • Region A lies above the curve and to the right of the \(y\)-axis.
    🔗
    🔗
  • Region B lies below the curve and to the right of the \(y\)-axis.
    🔗
    🔗
  • Region C lies above the curve and to the left of the \(y\)-axis.
    🔗
    🔗
  • Region D lies below the curve and to the left of the \(y\)-axis.
    🔗
    🔗
🔗
\(\begin{aligned} 2/x \amp \ge y \\ x \amp \ge 0 \end{aligned}\)
🔗
Answer.
\(\text{B}\)
🔗
🔗
🔗

Checkpoint 4.6.9.

Let \(f(x) = -4 x + 7\text{.}\) Find \(f^{-1}(x)\text{.}\)
🔗
\(f^{-1}(x) =\) .
🔗
Now for fun, verify that \((f\circ f^{-1})(x)=(f^{-1}\circ f)(x)=x\)
🔗
Answer.
\(\frac{-x}{4}+1.75\)
🔗
🔗
🔗

Checkpoint 4.6.10.

Let \(f(x) = 6+\sqrt{x-5}\text{.}\) Find \(f^{-1}(x)\text{.}\)
🔗
\(f^{-1}(x) =\) .
🔗
Now for fun, verify that \((f\circ f^{-1})(x)=(f^{-1}\circ f)(x)=x\)
🔗
Answer.
\(\left(x-6\right)^{2}+5\)
🔗
🔗
🔗

Checkpoint 4.6.11.

Find the inverse for each of the following functions.
🔗
\(f (x) = 10 x + 2\)
🔗
\(f^{-1}( x ) =\)
🔗
\(g(x) = 8 x^3 -1\)
🔗
\(g^{-1}( x ) =\)
🔗
\(\displaystyle h(x) = \frac{10}{x + 1}\)
🔗
\(h^{-1}( x ) =\)
🔗
\(j(x) = \sqrt[3]{x + 8}\)
🔗
\(j^{-1}( x ) =\)
🔗
Answer 1.
\(\frac{x-2}{10}\)
🔗
🔗
Answer 2.
\(\left(\frac{x+1}{8}\right)^{0.333333}\)
🔗
🔗
Answer 3.
\(\frac{10}{x}-1\)
🔗
🔗
Answer 4.
\(x^{3}-8\)
🔗
🔗
🔗

Checkpoint 4.6.12.

Determine whether the following function has an inverse. If it has an inverse, find its inverse function \(f^{-1}(x)\text{.}\) If it does not have an inverse, write “undefined” in the blank provided. You do not need to simplify your answer.
🔗
\(f(x)= x^2 + 5\)
🔗
\(f^{-1}(x)=\)
🔗
Answer.
\(\text{undefined}\)
🔗
🔗
🔗

Checkpoint 4.6.13.

Evaluate the following expressions. Your answers must be exact and in simplest form.
🔗
(a) \(\log_{17}(17^{7}) =\)
🔗
(b) \(\log_{2}(64) =\)
🔗
(c) \(\log_{6}(216) =\)
🔗
(d) \(\log_{9}(9^{4}) =\)
🔗
Answer 1.
\(7\)
🔗
🔗
Answer 2.
\(6\)
🔗
🔗
Answer 3.
\(3\)
🔗
🔗
Answer 4.
\(4\)
🔗
🔗
🔗

Checkpoint 4.6.14.

Enter a T or an F in each answer space below to indicate whether the corresponding statement is true or false.
🔗
1. \(\log_2(ab) = \log_2(a) + \log_2(b)\)
🔗
2. \(\ln(x-y) = \ln(x) - \ln(y)\)
🔗
3. \(\ln(a^b) = b \ln(a)\)
🔗
4. \(\frac {\ln(x)}{\ln(y)} = \ln(x) - \ln(y)\)
🔗
Answer 1.
\(\text{T}\)
🔗
🔗
Answer 2.
\(\text{F}\)
🔗
🔗
Answer 3.
\(\text{T}\)
🔗
🔗
Answer 4.
\(\text{F}\)
🔗
🔗
🔗

Checkpoint 4.6.15.

The domain of the function \(g(x)=\log_a(x^2-49)\) is
🔗
\((-\infty,\)\()\cup(\)\(,\infty)\)
🔗
Answer 1.
\(-7\)
🔗
🔗
Answer 2.
\(7\)
🔗
🔗
🔗

Checkpoint 4.6.16.

Use properties of the functions to match each of the following functions with its graph. Do not use your calculator.
🔗
1. \({\ln\mathopen{}\left(x-2\right)}\)
🔗
2. \({2+\ln\mathopen{}\left(x\right)}\)
🔗
3. \({\ln\mathopen{}\left(2-x\right)}\)
🔗
4. \({-\ln\mathopen{}\left(-x\right)}\)
🔗
5. \({\ln\mathopen{}\left(-x\right)}\)
🔗
Answer 1.
\(\text{D}\)
🔗
🔗
Answer 2.
\(\text{E}\)
🔗
🔗
Answer 3.
\(\text{B}\)
🔗
🔗
Answer 4.
\(\text{A}\)
🔗
🔗
Answer 5.
\(\text{C}\)
🔗
🔗
🔗

Checkpoint 4.6.17.

Checkpoint 4.6.18.

Which of the regions A, B, C, D belong to the solution set of the given system of inequalities.
🔗
The regions A, B, C, D are nonoverlapping regions bounded by the indicated lines.
🔗
\(\begin{array}{r@{}r@{}r@{}r} 3 x^2 \amp - y \amp \ge \amp 0 \\ 2 x \amp + 3 y \amp \le \amp 6 \end{array}\)
🔗
Answer.
\(\text{B}\)
🔗
🔗
🔗

Checkpoint 4.6.19.

Which of the regions A, B, C, D belong to the solution set of the given system of inequalities.
🔗
The regions A, B, C, D are nonoverlapping regions bounded by the indicated lines.
🔗
\(\begin{array}{r@{}r@{}r@{}r} 3 x \amp -2 y \amp \le \amp 0 \\ 2 x \amp + 3 y \amp \le \amp 6 \end{array}\)
🔗
Answer.
\(\text{C}\)
🔗
🔗
🔗

Checkpoint 4.6.20.

For each graph, determine the system of linear inequalities for which the solution set is shown on the graph.
🔗
1.
🔗
  • \(\displaystyle \begin{cases} y \ge 5 x \\ x \ge 4 y \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} y \ge 4 x \\ x \ge 5 y \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} y \le 5 x \\ x \ge 0 \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} y \le 5 x \\ x \le 4 y \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} y \ge 5 x \\ x \le 4 y \end{cases} \)
    🔗
    🔗
  • None of the above
    🔗
    🔗
🔗
2.
🔗
  • \(\displaystyle \begin{cases} y \le x \\ y \le 2 \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} y \le x + 2 \\ x \ge 2 \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} y \le - x - 2 \\ y \le 2 \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} y \le x + 2 \\ y \ge 2 \end{cases} \)
    🔗
    🔗
  • \(\displaystyle \begin{cases} x \le y - 2 \\ y \ge 2 \end{cases} \)
    🔗
    🔗
  • None of the above
    🔗
    🔗
🔗
Answer 1.
\(\text{Choice 4}\)
🔗
🔗
Answer 2.
\(\text{Choice 4}\)
🔗
🔗
🔗
🔗
PrevTopNext