4.5E: Describing Relationships with Rational Functions (Exercises)

Last updated
Save as PDF

Page ID: 99741

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \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}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$\newcommand{\longvect}{\overrightarrow}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

Section 4.5 Exercises

Part 1: Simplified Rational Functions

In exercises #1-4, (a) identify the undefined value(s) (b) Find the zero(s); (c) identify the end behavior; (d) match each equation form with one of the graphs without technological assistance.

1. $f(x)=\dfrac{x-4}{x+2}$

2. $g(x)=\dfrac{x^2-8x+16}{x+2}$

3. $h(x)=\dfrac{x-4}{x^2+4x+4}$

4. $k(x)=\dfrac{x^2-2x-8}{x^2-x-2}$

A. $The graph of a rational function with one vertical asympote, one x-intercept, and no horizontal asymptote.$ B. $The graph of a rational function with one vertical asympote, no x-intercepts, and a horizontal asymptote at y=0$

C. $The graph of a rational function with two vertical asympotes, two x-intercepts, and a horizontal asymptote at y=1$ D. $The graph of a rational function with one vertical asympote, one x-intercept, and a horizontal asymptote at y=1$

In exercises #5 -12, for the simplified rational function: (a) identify the undefined value(s) (b) Find the zero(s); (c) identify the end behavior (if a horizontal asymptote occurs, identify the horizontal asymptote); (d) use the given sign chart to graph the function without technological assistance.

5.$ f(x)=\dfrac{x-2}{x+1}$

$A sign chart for the graph of a rational function with one undefined value and one zero$

6. $ g(x)=\dfrac{x+5}{2x-1}$

$A sign chart for the graph of a rational function with one undefined value and one zero$

7. $ h(x)=\dfrac{-4}{x^2-4x+4 }$

$A sign chart for the graph of a rational function with two undefined values and no zeros$

8. $ k(x)=\dfrac{x^{2} -2x-3}{x-1}$

$A sign chart for the graph of a rational function with two undefined values and one zero$

9. $g\left(x\right)=\dfrac{x-2}{x^{2} -2x+1}$

$clipboard_ec859ef0f20117fcd5ab896112353a932.png$

Part 2: General Rational Functions

In exercises #10 -19, for the rational function: (a) identify the undefined value(s) and determine whether a vertical asymptote or hole occurs at the undefined value; (b) Find the zero(s); (c) identify the end behavior (if a horizontal asymptote occurs, identify the horizontal asymptote); (d) determine the sign of each interval; and (e) use this information to graph the function without technological assistance.

10. $p(x)=\dfrac{2x-3}{x+4}$

11. $q(x)=\dfrac{x-5}{3x-1}$

12. $s(x)=\dfrac{4}{x^2-4x+4 }$

13. $f(x)=\dfrac{x^{2} +x-6}{x^{2} -25}$

14. $g(x)=\dfrac{2x^{2} +7x-15}{3x^{2} -14x+15}$

15. $a(x)=\dfrac{x^{2} +2x-3}{x^{2} -1}$

16. $b(x)=\dfrac{x^{2} -x-6}{x^{2} -4}$

17. $k(x)=\dfrac{2x^{2} -3x-20}{x-5}$

18. $f(x)=\dfrac{3x^{2} -14x-5}{3x^{2} +8x-16}$

19. $n(x)=\dfrac{3x^{2} +4x-4}{x^{3} -4x^{2} }$

In exercises #19 -24, find a possible formula for a rational function whose graph is given below.

19. $The graph of a rational function with two vertical asympotes, one x-intercept, and a horizontal asymptote at y=0.$ 20. $The graph of a rational function with two vertical asympotes, one x-intercept, and a horizontal asymptote at y=0.$ 21. $The graph of a rational function with one vertical asympote, one x-intercept, and no horizontal asymptote.$

22. $The graph of a rational function with two vertical asympotes, two x-intercepts, and a horizontal asymptote at y=-2.$ 23. $The graph of a rational function with two vertical asympotes, one x-intercept, and a horizontal asymptote at y=2.$ 24. $The graph of a rational function with one vertical asympote, one x-intercept, a horizontal asymptote at y=1, and a hole at x=4.$

25. A scientist has a beaker containing 20 mL of a solution containing 20% acid. To dilute this, she adds pure water. The concentration in the beaker after adding $n$ mL of water is given by $C(n)=\dfrac{4}{20+n} $

a. Find the concentration if 10 mL of water has been added.
c. How many mL of water must be added to obtain a 4% solution?
d. What is the right end behavior as $n \to \infty$? What is the physical significance of this?

26. A gauss meter is a device for measuring the strength and polarity of magnetic fields. Suppose the following two measurements are made. At a distance of 8 feet from a magnet, the meter reads 2.3. At a distance of 6 feet, the meter reads 4.4. Then the meter reading $m(x)$ can modeled by the rational function $m(x)=\dfrac{134.9}{x^2-5.33$ where $x$ represents the distance in feet from a magnet.

How far must he go for the meter to reach 10? 100?
What is the right end behavior as $n \to \infty$? What is the physical significance of this?

27. The speed $s$ it takes to cover a distance of 20 miles is modeled by the function $s=\dfrac{20}{t}$. Find the vertical asymptote. What is the physical significance of the vertical asymptote?

Part 3: Rational Inequalities

In #28 - 31, solve the rational inequalities. Express your result in interval notation.

28. $ \dfrac{x+1}{x-3} \ge 0 $

29. $ \dfrac{x-2}{x^2-4x} < 0 $

30. $ \dfrac{1}{x-1} > 2 $

31. $ \dfrac{3}{x-2} \le \dfrac{2}{x} $

Answer

1. (a) $ f(x) $ has an undefined value at $x=-2$. A vertical asymptote occurs here. (b) $ f(x) $ has a zero at $x=4$ (c) As $ x \to \pm \infty$, $ y \to 1$. A horizontal asymptote occurs at $y=1 $. (d) Graph D

(d)

5. (a) $ f(x) $ has an undefined value at $x=-1$. A vertical asymptote occurs here. (b) $ f(x) $ has a zero at $x=2$ (c) As $ x \to \pm \infty$, $ y \to 1$. A horizontal asymptote occurs at $y=1 $. (d)

(d) $The graph of a rational function with a zero at x = 2, a vertical asymptote at x = -1, and a horizontal asymptote at y = 1$

10. (a) $ p(x) $ has an undefined value at $x=-4$. A vertical asymptote occurs here since $p$ is simplified. (b) $ p(x) $ has a zero at $x=\dfrac{2}{3}$ (c) As $ x \to \pm \infty$, $ y \to 2$. A horizontal asymptote occurs at $y=2 $.

(d) $a sign chart for the rational function$ (e) $The graph of a rational function with one vertical asymptote, one zero, and a horizontal asymptote at y = 2.$

15. (a) $ a(x) $ has undefined values at $x=1$ and $x=-1$. A vertical asymptote occurs at $x=-1$. A hole occurs at $x=1$. (b) $ a(x) $ has a zero at $x=-3$ (c) As $ x \to \pm \infty$, $ y \to 1$. A horizontal asymptote occurs at $y=1 $.

(d) $A sign chart for function f$ (e) $The graph of a rational function with one vertical asymptote, one hole, one zero, and a horizontal asymptote at y = 1.$

19. $q(x)=\dfrac{x-3}{(x+3)(x-4)}$

28. $ (-\infty,-1] \cup (3,\infty) $

Search

Text Color

Text Size

Margin Size

Font Type