Chapter 3, Problem 13RCC

a.

To determine

To define a rational function

Explanation of Solution

Given information:

The given statement is:

“What is a rational function”

A function that is written in the form $\text{r}\left(\text{x}\right)\text{=}\frac{\text{P}\left(\text{x}\right)}{\text{Q}\left(\text{x}\right)}$ where $\text{P}$ and $\text{Q}$ are polynomials is called a rational function. It is assumed that $\text{P}\left(\text{x}\right)$ and $\text{Q}\left(\text{x}\right)$ have no factors in common.

b.

To determine

To determine the meaning of the statement “ $\text{x=a}$ is a vertical asymptote of $\text{y=f}\left(\text{x}\right)$ ”

Explanation of Solution

Given information:

The given statement is:

“What does it mean to say that $\text{x=a}$ is a vertical asymptote of $\text{y=f}\left(\text{x}\right)$ ”

$\text{x=a}$ is a vertical asymptote of $\text{y=f}\left(\text{x}\right)$ means that $\text{y}$ approaches $\pm \infty$ as $\text{x}$ approaches $\text{a}$ from the right or left.

c.

To determine

To determine the process of location of a vertical asymptote.

Explanation of Solution

Given information:

The given statement is:

“How do you locate a vertical asymptote”

To locate a vertical asymptote , let $\text{r}$ be a rational function:

  $\text{r}\left(\text{x}\right)\text{=}\frac{{\text{a}}_{\text{n}}{\text{x}}^{\text{n}}{\text{+a}}_{\text{n}-\text{1}}{\text{x}}^{\text{n}-\text{1}}\text{+}\mathrm{...}{\text{+a}}_{\text{1}}{\text{x+a}}_{\text{0}}}{{\text{b}}_{\text{m}}{\text{x}}^{\text{m}}{\text{+b}}_{\text{m}-\text{1}}{\text{x}}^{\text{m}-\text{1}}\text{+}\mathrm{...}{\text{+b}}_{\text{1}}{\text{x+b}}_{\text{0}}}$

The lines $\text{x=a}$ , where $\text{a}$ is a zero of the denominator, are the vertical asymptotes of $\text{r}$ .

d.

To determine

To determine the meaning of the statement “ $\text{y=b}$ is a horizontal asymptote of $\text{y=f}\left(\text{x}\right)$ ”

Explanation of Solution

Given information:

The given statement is:

“What does it mean to say that $\text{y=b}$ is a horizontal asymptote of $\text{y=f}\left(\text{x}\right)$ ”

$\text{y=b}$ is a horizontal asymptote of $\text{y=f}\left(\text{x}\right)$ means that $\text{y}$ approaches $\text{b}$ as $\text{x}$ approaches $\pm \infty$ .

e.

To determine

To determine the process of location of a horizontal asymptote.

Explanation of Solution

Given information:

The given statement is:

“How do you locate a horizontal asymptote”

To locate a horizontal asymptote , let $\text{r}$ be a rational function:

  $\text{r}\left(\text{x}\right)\text{=}\frac{{\text{a}}_{\text{n}}{\text{x}}^{\text{n}}{\text{+a}}_{\text{n}-\text{1}}{\text{x}}^{\text{n}-\text{1}}\text{+}\mathrm{...}{\text{+a}}_{\text{1}}{\text{x+a}}_{\text{0}}}{{\text{b}}_{\text{m}}{\text{x}}^{\text{m}}{\text{+b}}_{\text{m}-\text{1}}{\text{x}}^{\text{m}-\text{1}}\text{+}\mathrm{...}{\text{+b}}_{\text{1}}{\text{x+b}}_{\text{0}}}$

1. If $\text{n}<\text{m}$ , then $\text{r}$ has horizontal asymptote $\text{y=0}$ .
2. If $\text{n}=\text{m}$ , then $\text{r}$ has a horizontal asymptote $\text{y=}\frac{{\text{a}}_{\text{n}}}{{\text{b}}_{\text{m}}}$ .
3. If $\text{n}>\text{m}$ , then $\text{r}$has no horizontal asymptote.

f.

To determine

To write the steps to be followed to sketch the graph of a rational function by hand.

Explanation of Solution

Given information:

The given statement is:

“What steps do you follow to sketch the graph of a rational function by hand”

Follow the given steps to sketch the graph of a rational function by hand:

1. Factor the numerator and denominator.
2. By determining the zeroes of the numerator find the x-intercepts and from the value of the function at $\text{x}=\text{0}$, find the y-intercept.
3. By determining the zeroes on the denominator find the vertical asymptotes and then see whether $\text{y}\to \infty$ or $\text{y}\to -\infty$on each side of each vertical asymptotes by using test values.
4. Find the horizontal asymptote.
5. Graph the information provided in the above steps. Then plot as many additional points as needed to fill in the rest of the graph.

g.

To determine

To determine the conditions under which a rational function have a slant asymptote and to find the slant asymptote.

Explanation of Solution

Given information:

The given statement is:

“Under what circumstances does a rational function have a slant asymptote? If one exists, how do you find it?”

If $\text{r}\left(\text{x}\right)=\frac{\text{P}\left(\text{x}\right)}{\text{Q}\left(\text{x}\right)}$ is a rational function in which the degree of numerator is one more than the degree of denominator, by division algorithm the function can be expressed as $\text{r}\left(\text{x}\right)=\text{ax+b+}\frac{\text{R}\left(\text{x}\right)}{\text{Q}\left(\text{x}\right)}$ where the degree of $\text{R}$ is less than the degree of $\text{Q}$ and $\text{a}\ne \text{0}$ . It means that as $\text{x}\to \pm \infty ,\frac{\text{R}\left(\text{x}\right)}{\text{Q}\left(\text{x}\right)}\to 0$ , and for large values of $\left|\text{x}\right|$ the graph of $\text{y=r}\left(\text{x}\right)$ approaches the graph of the line $\text{y}=\text{ax}+\text{b}$ . Under these circumstances, $\text{y}=\text{ax}+\text{b}$ is a slant asymptote.

To find a slant asymptote, compare the degree of the numerator and the denominator. If the degree of the numerator is one more than the degree of the denominator, it is a slant asymptote.

h.

To determine

To determine the end behavior of a rational function.

Explanation of Solution

Given information:

The given statement is:

“How do you determine the end behavior of a rational function?”

To determine the end behavior of a rational function, see if $\text{x}\to \pm \infty$ .

1. When the degree of the numerator is more than the degree of the denominator, the end behavior is a slant asymptote.
2. When the degree of the numerator and the denominator is the same, the end behavior is a horizontal asymptote.
3. When the denominator is zero, the end behavior is a vertical asymptote.
4. If the graph is close to $\text{y}={\text{x}}^2$
5. , the end behavior is like that of a parabola.