Nelson Functions 11
Nelson Functions 11
1st Edition
Chris Kirkpatrick, Marian Small
ISBN: 9780176332037
Textbook solutions

All Solutions

Section 2-5: Exploring Graphs of Rational Functions

Exercise 1
Step 1
1 of 2
A rational function with a horizontal graph except for two holes is $f(x)=dfrac{5x^2+5x}{x^2+x}$
since

$$
begin{align*}
f(x)&=dfrac{5x^2+5x}{x^2+x}
\\&=
dfrac{5x(x+1)}{x(x+1)}
\\&=
dfrac{5cancel{x(x+1)}}{cancel{x(x+1)}}
\\&=
5
.end{align*}
$$

The graph of $f(x)=5$ is a horizontal line passing through $(0,5)$ where $x={0,1}$ are not defined.

See graph below.

Exercise scan

Result
2 of 2
$f(x)=dfrac{5x^2+5x}{x^2+x}$ is a rational function example with a horizontal graph with $2$ holes
Exercise 2
Step 1
1 of 2
A rational function with a graph entirely above the $x$-axis and has a single vertical asymptote is $f(x)=dfrac{5x+5}{x+1}$
since

$$
begin{align*}
f(x)&=dfrac{5(x+1)}{x+1}
\\
f(x)&=dfrac{5(cancel{x+1})}{cancel{x+1}}
\\
f(x)&=5
.end{align*}
$$

The graph of $f(x)=5$ is a line lying entirely above the $x$-axis and has the single vertical asymptote, $x=-1.$

See graph below.

Exercise scan

Result
2 of 2
$f(x)=dfrac{5x+5}{x+1}$ is a rational function example with a graph entirely above the $x$-axis and with a single vertical asymptote
Exercise 3
Step 1
1 of 2
The rational function, $f(x)=dfrac{2x^2}{(x+1)(x-1)}
,$
has the horizontal asymptote $y=2$ and two vertical asymptiotes, $x=1$ and $x=-1$.

See graph below.

Exercise scan

Result
2 of 2
$f(x)=dfrac{2x^2}{(x+1)(x-1)}$ is a rational function example with a horizontal asymptote, $y=2,$ and two vertical asymptotes
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