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Precalculus Review Materials

Jane Butterfield, Gary MacGillivray

Section 3.6 Exercises

These are the exercises in Module 3.
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Note that many of these exercises contain images, which are by default not loaded. Make sure to click ’Make interactive’ on each in order to load the images.
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Checkpoint 3.6.1.

Solve the following inequality. Write the answer in interval notation.
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\(\frac{x-6}{x+19}\geq 0\)
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Answer:
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Answer.
\(\left(-\infty ,-19\right)\cup \left[6,\infty \right)\)
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Checkpoint 3.6.2.

Solve the following inequality. Enter the answer in interval notation.
\begin{equation*} \frac{4 -x}{x-8} \ge 0 \end{equation*}
Answer:
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Note: If the answer includes more than one interval, write the intervals separated by the “union” symbol, U. If needed, enter \(\infty\) as infinity and \(-\infty\) as -infinity.
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Answer.
\(\left[4,8\right)\)
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Checkpoint 3.6.3.

Solve the following inequality. Write the answer in interval notation.
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\({\frac{x+3}{x-4}} \lt 1\)
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Solution set (in interval notation):
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Answer.
\(\left(-\infty ,4\right)\)
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Checkpoint 3.6.4.

Solve the following inequality. Write the answer in interval notation.
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\(\frac{1}{(v - 3)} \leq \frac{1}{(v + 10)}\)
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Answer:
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Answer.
\(\left(-10,3\right)\)
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Checkpoint 3.6.5.

The expression
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\begin{equation*} \frac{2^52^32^{-4}}{\sqrt{4^4}2^32^{-3}} \end{equation*}
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equals \(2^n\) where \(n\) is:
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Answer.
\(0\)
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Solution.
\(\begin{aligned} \amp \phantom{\mathrel{{}={}}}\frac{2^{5}2^{3}2^{-4} }{ \sqrt{4^{4}}2^{3}2^{-3}} \\ \amp =\frac{2^{5+3-4} }{ ((2^2)^4)^{\frac{1}{2}}2^{3-3}} \\ \amp =\frac{2^{4} }{ 2^{4}2^{0}} \\ \amp =2^{4 - (4 + 0) } \\ \amp =2^{0} \end{aligned}\)
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Checkpoint 3.6.6.

The expression
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\(\left(y^{\frac{1}{4}}\right)^{\frac{3}{5}}\)
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equals \(y^r\) where \(r\text{,}\) the exponent of \(y\text{,}\) is:
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Answer.
\({\frac{3}{20}}\)
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Checkpoint 3.6.7.

The expression
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\((3 x^3 y^{\frac{-6}{7}})^5(4 y^2)^{\frac{1}{2}}\)
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equals \(n x^r y^t\) where
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\(n\text{,}\) the coefficient, is:
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\(r\text{,}\) the exponent of \(x\text{,}\) is:
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\(t\text{,}\) the exponent of \(y\text{,}\) is:
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Answer 1.
\(486\)
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Answer 2.
\(15\)
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Answer 3.
\(-{\frac{23}{7}}\)
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Checkpoint 3.6.8.

The expression
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\((c^3d^4)^{\frac{-1}{5}}\)
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equals \(\frac{1}{(c^r d^s)}\) where
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\(r\text{,}\) the exponent of \(c\text{,}\) is:
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\(s\text{,}\) the exponent of \(d\text{,}\) is:
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Answer 1.
\({\frac{3}{5}}\)
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Answer 2.
\({\frac{4}{5}}\)
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Checkpoint 3.6.9.

Find coefficients \(a\text{,}\) \(h\) and \(k\) so
\begin{equation*} y=a(x-h)^2+k \end{equation*}
is an equation for the quadratic function graphed above.
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\(a =\), \(h =\), \(k =\).
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Answer 1.
\(-1\)
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Answer 2.
\(-1\)
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Answer 3.
\(-6\)
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Checkpoint 3.6.10.

The equation of the graphed function is
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Hint.
Look for evidence that will help eliminate wrong answers.
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This graph has a maximum (highest \(y\) value), are there any answers you can eliminate because they don’t have maximums?
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If more than one answer is left, can you eliminate any because you can see a point that is on the graph but does not satisfy the equation in that answer?
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Answer.
\(\text{Choice 3}\)
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Solution.
A look at the graph gives us the following information.
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  1. The answer cannot be \(y=5+x^2\) since this graph has a maximum at \((0,5)\) and \(y=5+x^2\) has no maximum as \(x\) increases.
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  2. The answer cannot be \(y=x^2-5\) for two reasons.
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First, when \(x=0\) we get \(y=x^2-5=-5\) and we know from the graph than when \(x=0, y=5\)
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Second, this graph has a maximum at \((0,5)\) and \(y=x^2-5\) has no maximum as \(x\) increases.
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  1. The answer cannot be \(y=-x^2\) since when \(x=0, y=-x^2=0\) and we know from the graph that when \(x=0, y=5\)
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To make sure we might want to check the remaining answer, \(y=5-x^2\text{.}\) It has the correct maximum at \((0,5\) and the graph looks like a parabola. If we were certain that the points at which the function corsses the \(x\)-axis were \((-\sqrt{5},0)\) and \((\sqrt{5},0)\) then we could be more sure, since just as the equation of a line can be found from knowing two points, the equation of a parabola can be found from knowing three points.
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Checkpoint 3.6.11.

Use properties of the functions to match each of the following functions with its graph. Do not use your calculator.
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1. \({-x^{2}+2}\)
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2. \({\frac{-x\mathopen{}\left(3-x\right)^{2}}{3}}\)
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3. \({\frac{x^{3}}{3}-\frac{x^{2}}{3}-3x+0.5}\)
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4. \({\frac{-x^{3}}{3}+\frac{x^{2}}{3}+3x+0.5}\)
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Answer 1.
\(\text{B}\)
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Answer 2.
\(\text{D}\)
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Answer 3.
\(\text{A}\)
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Answer 4.
\(\text{C}\)
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Checkpoint 3.6.12.

What is the smallest degree polynomial which could have the following graph?
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Smallest possible degree =
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Answer.
\(6\)
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Checkpoint 3.6.13.

Tell whether the following ordered pairs are solutions to the given inequality.
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\(y \le 4\)
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  1. \((-8,4)\)
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  2. \((0,0)\)
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  3. \((-5,-6)\)
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  4. \((0,8)\)
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Answer 1.
\(\text{is a solution}\)
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Answer 2.
\(\text{is a solution}\)
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Answer 3.
\(\text{is a solution}\)
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Answer 4.
\(\text{is not a solution}\)
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Solution.
We substitute \(y\) values from each ordered pair into the inequality \(y \le 4\) and see whether the inequality is true.
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  1. The ordered pair is \((-8,4)\text{.}\)
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    \(\begin{aligned}[t] y \amp \stackrel{?}{\le} 4 \\ 4 \amp \stackrel{?}{\le} 4 \end{aligned}\)
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    Since this inequality is true, \((-8,4)\) is a solution to the inequality.
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  2. The ordered pair is \((0,0)\text{.}\)
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    \(\begin{aligned}[t] y \amp \stackrel{?}{\le} 4 \\ 0 \amp \stackrel{?}{\le} 4 \end{aligned}\)
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    Since this inequality is true, \((0,0)\) is a solution to the inequality.
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  3. The ordered pair is \((-5,-6)\text{.}\)
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    \(\begin{aligned}[t] y \amp \stackrel{?}{\le} 4 \\ -6 \amp \stackrel{?}{\le} 4 \end{aligned}\)
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    Since this inequality is true, \((-5,-6)\) is a solution to the inequality.
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  4. The ordered pair is \((0,8)\text{.}\)
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    \(\begin{aligned}[t] y \amp \stackrel{?}{\le} 4 \\ 8 \amp \stackrel{?}{\le} 4 \end{aligned}\)
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    Since this inequality is not true, \((0,8)\) is not a solution to the inequality.
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Checkpoint 3.6.14.

This exercise is not currently working as expected, and was removed August 2023.
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Checkpoint 3.6.15.

Use properties of the functions to match each of the following functions with its graph. Do not use your calculator.
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1. \(y +x \ge -3\)
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2. \(x \le 2 \text{ and } y \ge 4\)
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3. \(x+y \le 2\)
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4. \(y \le -2\)
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Answer 1.
\(\text{B}\)
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Answer 2.
\(\text{A}\)
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Answer 3.
\(\text{D}\)
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Answer 4.
\(\text{C}\)
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Checkpoint 3.6.16.

This exercise is not currently working as expected, and was removed August 2023.
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Checkpoint 3.6.17.

Checkpoint 3.6.18.

Find the exponential function \(f(x)=a^x\) whose graph goes through the point \((2,16)\text{.}\)
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\(a=\) .
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Answer.
\(4\)
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Checkpoint 3.6.19.

Write the given exponential function in the form \(ab^t\text{.}\)
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\(Q=\frac{165}{3^t}\) has:
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\(a =\)
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\(b =\)
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Answer 1.
\(165\)
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Answer 2.
\({\frac{1}{3}}\)
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Checkpoint 3.6.20.

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