Chapter 4, Problem 24RE

To determine

To analyze: the given rational function.

Answer to Problem 24RE

  

Explanation of Solution

Given:

  $R\left(x\right)=\frac{4-x}{x}$

Calculation:

Let us consider the following rational function,

  $R\left(x\right)=\frac{4-x}{x}$

analyze the graph of the given function.

For this follow the following steps:

Step $1$ :

Determination the domain of the function. Domain of a function is defined as the set of values of $x$ for which the function is defined.

So, let us first find the values of $x$ for which $R$ is not defined.

  $R$ is not defined when the denominator is zero. So, find the value of $x$ for which the denominator is zero.

  $x=0$

So, at $x=0$ , $R$ is not defined.

Therefore, the domain of the function is $\left\{x|x\ne 0\right\}$

Step $2$ :

Write $R$ in the lowest terms.

Here, since there is no common factor between the numerator and the denominator, $R$ is in the lowest terms.

Step $3$ :

Next, approximate $x$ and the $y$ intercepts of the graph.

  $x-\mathrm{int}ercept$ of the graph is found by determining the zeros of the numerator.

So,

  $\begin{array}{l}4-x=0\\ x=4\end{array}$

Since there is only one zero of the numerator, there is only one $x-\mathrm{int}ercept$ at $x=4$ .

Also, since $x=0$ is not in the domain of $R$ , the graph doesn’t have a $y-\mathrm{int}ercept$ .

Step $4$ :

Determine whether the function is odd, even or neither.

A function is even if,

  $f\left(x\right)=f\left(-x\right)$

Also, a function is odd if,

  $f\left(-x\right)=-f\left(x\right)$

So, let us first find the values of $f\left(-x\right)$

  $\begin{array}{l}f\left(-x\right)=\frac{4-\left(-x\right)}{\left(-x\right)}\\ =\frac{4+x}{-x}\\ =-\left(\frac{4+x}{x}\right)\\ -f\left(x\right)=-\left(\frac{4+x}{x}\right)\\ =\frac{-4+x}{x}\\ =\frac{x-4}{x}\\ So,f\left(-x\right)\ne -f\left(x\right)\end{array}$

Therefore, the function is not odd.

Also,

  $f\left(-x\right)\ne f\left(x\right)$

Therefore, the function is not even

Therefore, the function is neither odd nor even.

So, the graph is neither symmetric about $y-axis$ nor by origin.

Step $5$ :

Find the vertical asymptotes.

The vertical asymptotes of a rational function are located at the real zeros of the denominator.

As found above, the real zero of the denominator here is $0$

So, the vertical asymptote of the function lies at $x=0$

Step $6$ :

Find the horizontal or oblique asymptotes. Also, locate the points where the asymptote intersects the graph.

  $R\left(x\right)=\frac{4-x}{x}$

Here, the degree of the numerator and the denominator is same. Therefore, this is an improper rational function. So, to find the horizontal asymptotes, use the long division method.

  $x\begin{array}{c}\hfill -1\\ \hfill \overline{)\begin{array}{l}4-x\\ \frac{-x|}{4}\end{array}}\end{array}$

So,

  $\frac{4-x}{x}=-1+\frac{4}{x}$

Here, as $x\to -\infty$ or $x\to \infty$ , the value of the function approaches $-1$

Therefore, the horizontal asymptote of the function lies at $y=-1$

To determine where the graph intersects the horizontal asymptote, solve $R\left(x\right)=-1$

So,

  $\begin{array}{l}\frac{4-x}{x}=-1\\ 4-x=-x\end{array}$

This is not defined.

So, the graph doesn’t intersect the horizontal asymptote.

Step $7$ ;

Graph $R$ using graphing utility.

  

Step $8$ :

Complete the graph using all the above information.

  

Conclusion:

Therefore,the given polynomial function is analyzed.