College Algebra

Ron Larson

Chapter 4

Rational Functions and Conics - all with Video Answers

Educators

+ 2 more educators

Section 1

Rational Functions and Asymptotes

Problem 1

Functions of the form $f(x)=N(x) / D(x),$ where $N(x)$ and $D(x)$ are polynomials and $D(x)$ is not the zero polynomial, are called _________ _________.

SL

Steven La

SUNY at Binghamton

Problem 2

When $f(x) \rightarrow \pm \infty$ as $x \rightarrow a$ from the left or the right, $x=a$ is a _______ ___________ of the graph of $f$

JW

Julie Wyman

Numerade Educator

Problem 3

When $f(x) \rightarrow b$ as $x \rightarrow \pm \infty, y=b$ is a __________ _________ of the graph of $f$

Zain Haider

Numerade Educator

Problem 4

The graph of $f(x)=1 / x$ is called a _________.

Yujie Wang

College of San Mateo

Problem 5

find the domain of the function, and discuss the behavior of $f$ near any excluded $x$ -values.
$$
f(x)=\frac{1}{x-1}
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 6

find the domain of the function, and discuss the behavior of $f$ near any excluded $x$ -values.
$$
f(x)=\frac{4}{x+3}
$$

Yujie Wang

College of San Mateo

Problem 7

find the domain of the function, and discuss the behavior of $f$ near any excluded $x$ -values.
$$
f(x)=\frac{5 x}{x+2}
$$

Christy Galilei

Christy Galilei

Numerade Educator

Problem 8

find the domain of the function, and discuss the behavior of $f$ near any excluded $x$ -values.
$$
f(x)=\frac{6 x}{x-3}
$$

Yujie Wang

College of San Mateo

Problem 9

find the domain of the function, and discuss the behavior of $f$ near any excluded $x$ -values.
$$
f(x)=\frac{3 x^{2}}{x^{2}-1}
$$

Yujie Wang

College of San Mateo

Problem 10

find the domain of the function, and discuss the behavior of $f$ near any excluded $x$ -values.
$$
f(x)=\frac{2 x}{x^{2}-4}
$$

Yujie Wang

College of San Mateo

Problem 11

find the domain of the function, and discuss the behavior of $f$ near any excluded $x$ -values.
$$
f(x)=\frac{x^{2}+3 x+2}{x^{2}-2 x+1}
$$

Yujie Wang

College of San Mateo

Problem 12

find the domain of the function, and discuss the behavior of $f$ near any excluded $x$ -values.
$$
f(x)=\frac{x^{2}+x-20}{x^{2}+8 x+16}
$$

Allison Knapp

Numerade Educator

Problem 13

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{4}{x^{2}}
$$

Narayan Hari

Numerade Educator

Problem 14

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{1}{(x-2)^{3}}
$$

Narayan Hari

Numerade Educator

Problem 15

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{5+x}{5-x}
$$

Yujie Wang

College of San Mateo

Problem 16

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{3-7 x}{3+2 x}
$$

Narayan Hari

Numerade Educator

Problem 17

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{x^{3}}{x^{2}-1}
$$

Yujie Wang

College of San Mateo

Problem 18

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{2 x^{2}}{x+1}
$$

Yujie Wang

College of San Mateo

Problem 19

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{3 x^{2}+1}{x^{2}+x+9}
$$

Yujie Wang

College of San Mateo

Problem 20

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{3 x^{2}+x-5}{x^{2}+1}
$$

Yujie Wang

College of San Mateo

Problem 21

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{x-4}{x^{2}-16}
$$

Yujie Wang

College of San Mateo

Problem 22

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{x+3}{x^{2}-9}
$$

Yujie Wang

College of San Mateo

Problem 23

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{x^{2}-1}{x^{2}-2 x-3}
$$

Yujie Wang

College of San Mateo

Problem 24

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{x^{2}-4}{x^{2}-3 x+2}
$$

Yujie Wang

College of San Mateo

Problem 25

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{x^{2}-3 x-4}{2 x^{2}+x-1}
$$

Narayan Hari

Numerade Educator

Problem 26

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{x^{2}+x-2}{2 x^{2}+5 x+2}
$$

Yujie Wang

College of San Mateo

Problem 27

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{6 x^{2}+5 x-6}{3 x^{2}-8 x+4}
$$

Yujie Wang

College of San Mateo

Problem 28

find all vertical and horizontal asymptotes of the graph of the function.
$$
f(x)=\frac{6 x^{2}-11 x+3}{6 x^{2}-7 x-3}
$$

Yujie Wang

College of San Mateo

Problem 29

match the rational function with its graph. [The graphs are labeled (a)-(h).]
(graph cant copy)
$$
f(x)=\frac{4}{x+5}
$$

Yujie Wang

College of San Mateo

Problem 30

match the rational function with its graph. [The graphs are labeled (a)-(h).]
(graph cant copy)
$$
f(x)=\frac{5}{x-2}
$$

Yujie Wang

College of San Mateo

Problem 31

match the rational function with its graph. [The graphs are labeled (a)-(h).]
(graph cant copy)
$$
f(x)=\frac{x-1}{x-4}
$$

Yujie Wang

College of San Mateo

Problem 32

match the rational function with its graph. [The graphs are labeled (a)-(h).]
(graph cant copy)
$$
f(x)=-\frac{x+2}{x+4}
$$

Yujie Wang

College of San Mateo

Problem 33

match the rational function with its graph. [The graphs are labeled (a)-(h).]
(graph cant copy)
$$
f(x)=\frac{2 x^{2}}{x^{2}-1}
$$

Yujie Wang

College of San Mateo

Problem 34

match the rational function with its graph. [The graphs are labeled (a)-(h).]
(graph cant copy)
$$
f(x)=\frac{-2 x}{x^{2}-1}
$$

Yujie Wang

College of San Mateo

Problem 35

match the rational function with its graph. [The graphs are labeled (a)-(h).]
(graph cant copy)
$$
f(x)=\frac{3}{(x-2)^{2}}
$$

Yujie Wang

College of San Mateo

Problem 36

match the rational function with its graph. [The graphs are labeled (a)-(h).]
(graph cant copy)
$$
f(x)=\frac{3 x}{(x+2)^{2}}
$$

Yujie Wang

College of San Mateo

Problem 37

(a) determine the domains of $f$ and $g,$ (b) simplify $f$ and find any vertical asymptotes of $f,$ (c) complete the table, and (d) explain how the two functions differ.
$$
f(x)=\frac{x^{2}-4}{x+2}, \quad g(x)=x-2
$$

Yujie Wang

College of San Mateo

Problem 38

(a) determine the domains of $f$ and $g,$ (b) simplify $f$ and find any vertical asymptotes of $f,$ (c) complete the table, and (d) explain how the two functions differ.
$$
f(x)=\frac{x^{2}(x+3)}{x^{2}+3 x}, \quad g(x)=x
$$
$$
\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & & & & & & & \\ \hline g(x) & & & & & & & \\ \hline \end{array}
$$

Christee Joesten

Christee Joesten

Numerade Educator

Problem 39

(a) determine the domains of $f$ and $g,$ (b) simplify $f$ and find any vertical asymptotes of $f,$ (c) complete the table, and (d) explain how the two functions differ.
$$
f(x)=\frac{2 x-1}{2 x^{2}-x}, \quad g(x)=\frac{1}{x}
$$
$$
\begin{array}{|l|l|l|l|l|l|l|l|} \hline x & -1 & -0.5 & 0 & 0.5 & 2 & 3 & 4 \\ \hline f(x) & & & & & & & \\ \hline g(x) & & & & & & & \\ \hline \end{array}
$$

Christee Joesten

Christee Joesten

Numerade Educator

Problem 40

(a) determine the domains of $f$ and $g,$ (b) simplify $f$ and find any vertical asymptotes of $f,$ (c) complete the table, and (d) explain how the two functions differ.
$$
f(x)=\frac{2 x-8}{x^{2}-9 x+20}, \quad g(x)=\frac{2}{x-5}
$$

Christee Joesten

Christee Joesten

Numerade Educator

Problem 41

The cost $C$ (in dollars) of supplying recycling bins to $p \%$ of the population of a rural township is given by
$$C=\frac{25,000 p}{100-p}, \quad 0 \leq p<100$$
(a) Use a graphing utility to graph the cost function.
(b) Find the costs of supplying bins to $15 \%, 50 \%,$ and $90 \%$ of the population.
(c) According to this model, would it be possible to supply bins to $100 \%$ of the residents? Explain.

Michael Anderson

Michael Anderson

Numerade Educator

Problem 42

Pollution The cost $C$ (in millions of dollars) of removing $p \%$ of the industrial and municipal pollutants discharged into a river is given by
$C=\frac{225 p}{100-p}, \quad 0 \leq p<100$
(a) Use a graphing utility to graph the cost function.
(b) Find the costs of removing $10 \%, 40 \%,$ and $75 \%$ of the pollutants.
(c) According to this model, would it be possible to remove $100 \%$ of the pollutants? Explain.

Michael Anderson

Michael Anderson

Numerade Educator

Problem 43

Population Growth The game commission introduces 100 deer into newly acquired state game lands. The population $N$ of the herd is modeled by
$$N=\frac{20(5+3 t)}{1+0.04 t}, \quad t \geq 0$$
where $t$ is the time in years.
(a) Use a graphing utility to graph this model.
(b) Find the populations when $t=5, t=10,$ and $t=25$
(c) What is the limiting size of the herd as time increases?

Chris Trentman

Numerade Educator

Problem 44

Food Consumption $A$ biology class performs an experiment comparing the quantity of food consumed by a certain kind of moth with the quantity supplied. The model for the experimental data is
$$y=\frac{1.568 x-0.001}{6.360 x+1}, \quad x>0$$
where $x$ is the quantity (in milligrams) of food supplied and $y$ is the quantity (in milligrams) of food consumed.
(a) Use a graphing utility to graph this model.
(b) At what level of consumption will the moth become satiated?

Ma. Theresa Alin

Ma. Theresa Alin

Numerade Educator

Problem 45

Human Memory Model Psychologists have developed mathematical models to predict memory performance as a function of the number of trials $n$ of a certain task. Consider the learning curve
$$P=\frac{0.5+0.9(n-1)}{1+0.9(n-1)}, \quad n>0$$
where $P$ is the fraction of correct responses after $n$ trials.
(a) Complete the table for this model. What does it suggest?
$$
\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline P & & & & & & & & & & \\ \hline \end{array}
$$

Sandra Lundell

Numerade Educator

Problem 46

Data Analysis: Physics Experiment Consider a physics laboratory experiment designed to determine an unknown mass. A flexible metal meter stick is clamped to a table with 50 centimeters overhanging the edge (see figure). Known masses $M$ ranging from 200 grams to 2000 grams are attached to the end of the meter stick. For each mass, the meter stick is displaced vertically and then allowed to oscillate. The average time $t$ (in seconds) of one oscillation for each mass is recorded in the table.
$$
\begin{array}{|c|c|} \hline \text { Wrose } & \text { Thines } \\ \hline 200 & 0.450 \\ 400 & 0.597 \\ 600 & 0.712 \\ 800 & 0.831 \\ 1000 & 0.906 \\ 1200 & 1.003 \\ 1400 & 1.088 \\ 1600 & 1.168 \\ 1800 & 1.218 \\ 2000 & 1.338 \\ \hline \end{array}
$$
A model for the data that can be used to predict the time of one oscillation is
$$t=\frac{38 M+16,965}{10(M+5000)}$$
(a) Use this model to create a table showing the predicted time for each of the masses shown in the table.
(b) Compare the predicted times with the experimental times. What can you conclude?
(c) Use the model to approximate the mass of an object for which $t=1.056$ seconds.

Joseph Palsic

Numerade Educator

Problem 47

determine whether the statement is true or false. Justify your answer.
The graph of a polynomial function can have infinitely many vertical asymptotes.

Narayan Hari

Numerade Educator

Problem 48

determine whether the statement is true or false. Justify your answer.
$f(x)=x^{3}-2 x^{2}-5 x+6$ is a rational function.

Yujie Wang

College of San Mateo

Problem 49

(a) determine the value that the function $f$ approaches as the magnitude of $x$ increases. Is $f(x)$ greater than or less than this functional value when (b) $x$ is positive and large in magnitude and (c) $x$ is negative and large in magnitude?
$$
f(x)=4-\frac{1}{x}
$$

Yujie Wang

College of San Mateo

Problem 50

(a) determine the value that the function $f$ approaches as the magnitude of $x$ increases. Is $f(x)$ greater than or less than this functional value when (b) $x$ is positive and large in magnitude and (c) $x$ is negative and large in magnitude?
$$
f(x)=2+\frac{1}{x-3}
$$

Yujie Wang

College of San Mateo

Problem 51

(a) determine the value that the function $f$ approaches as the magnitude of $x$ increases. Is $f(x)$ greater than or less than this functional value when (b) $x$ is positive and large in magnitude and (c) $x$ is negative and large in magnitude?
$$
f(x)=\frac{2 x-1}{x-3}
$$

Yujie Wang

College of San Mateo

Problem 52

(a) determine the value that the function $f$ approaches as the magnitude of $x$ increases. Is $f(x)$ greater than or less than this functional value when (b) $x$ is positive and large in magnitude and (c) $x$ is negative and large in magnitude?
$$
f(x)=\frac{2 x-1}{x^{2}+1}
$$

Yujie Wang

College of San Mateo

Problem 53

write a rational function $f$ that has the specified characteristics. (There are many correct answers.)
Vertical asymptote: None
Horizontal asymptote: $y=2$

Yujie Wang

College of San Mateo

Problem 54

write a rational function $f$ that has the specified characteristics. (There are many correct answers.)
Vertical asymptotes: $x=-2, x=1$
Horizontal asymptote: None

Yujie Wang

College of San Mateo

Problem 55

Think About It Give an example of a rational function whose domain is the set of all real numbers. Give an example of a rational function whose domain is the set of all real numbers except $x=15$

Yujie Wang

College of San Mateo

Problem 56

The graph of a rational function
$$f(x)=\frac{N(x)}{D(x)}$$
is shown below. Determine which of the statements about the function is false. Justify your answer.
(a) When $x=1, D(x)=0$
(b) The degrees of $N(x)$ and $D(x)$ are equal.
(c) The ratio of the leading coefficients of $N(x)$ and $D(x)$ is 1

Allison Knapp

Numerade Educator