Chapter 2.6, Problem 7E

Solution

To Describe:

The series of transformations that will change the reciprocal function $g\left(x\right)=\frac{1}{x}$ into the given function $f\left(x\right)$ .

To identify and describe behavior using limits at:

The horizontal and vertical asymptotes.

To Graph:

The function $f\left(x\right)$ .

  $f\left(x\right)=2-7g\left(x+3\right)$ . The graph of $f$ is the graph of the reciprocal function shifted $3$ units to the left, then scaled vertically by a factor of $7$ , then reflected across the $x$ -axis, and finally translated $2$ units up.

The vertical asymptote is $x=-3$ and the horizontal asymptote is $y=2$ . $\underset{x\to -3^{-}}{\lim }f\left(x\right)\to \infty$ , $\underset{x\to -3^{+}}{\lim }f\left(x\right)\to -\infty$ , and $\underset{x\to \pm \infty }{\lim }f\left(x\right)=2$ describe the behavior at these asymptotes.

  

Given:

  $f\left(x\right)=\frac{2x-2}{x+3}$ .

Concepts Used:

The reciprocal function $g\left(x\right)=\frac{1}{x}$ has horizontal asymptote at $y=0$ and vertical asymptote at $x=0$ . Also, $\underset{x\to \pm \infty }{\lim }g\left(x\right)=0$ , and $\underset{x\to 0^{\pm }}{\lim }g\left(x\right)\to \pm \infty$ .

If the reciprocal function $g\left(x\right)$ is modified to the form $b+kg\left(x-a\right)$ where $a\text{, }b\text{, and }k$ are constants, then the vertical asymptote shifts to $x=a$ and the horizontal asymptote is shifted to $y=b$ . Also, $\underset{x\to \pm \infty }{\lim }b+kg\left(x-a\right)=b$ , and $\underset{x\to a^{\pm }}{\lim }b+kg\left(x-a\right)\to k\times \pm \infty$ .

The behavior of a function $f$ at a vertical asymptote $x=c$ can be determined using the one sided limits at the point $x=c$ . The behavior at the horizontal asymptote is same as the end behavior of a function. The end behavior of a function $f\left(x\right)$ is indicated by the values of the limits $\underset{x\to -\infty }{\lim }f\left(x\right)$ and $\underset{x\to \infty }{\lim }f\left(x\right)$ .

Calculations:

Rewrite $f\left(x\right)$ in terms of the reciprocal function $g$ .

  $\begin{array}{c}f\left(x\right)=\frac{2x-1}{x+3}\\ =\frac{2\left(x+3-3\right)-1}{x+3}\\ =\frac{2\left(x+3\right)-2\left(3\right)-1}{x+3}\\ =\frac{2\left(x+3\right)-6-1}{x+3}\\ =\frac{2\left(x+3\right)-7}{x+3}\\ =\frac{2\left(x+3\right)}{x+3}-\frac{7}{x+3}\\ =2-\frac{7}{x+3}\\ =2-7g\left(x+3\right)\end{array}$

Thus, $f\left(x\right)=2-7g\left(x+3\right)$ . The graph of $f$ is the graph of the reciprocal function shifted $3$ units to the left, then scaled vertically by a factor of $7$ , then reflected across the $x$ -axis, and finally translated $2$ units up.

Compare $f\left(x\right)=2-7g\left(x+3\right)$ with $b+kg\left(x-a\right)$ and obtain the vertical asymptote $x=-3$ and the horizontal asymptote $y=2$ .

Using limits, describe the behavior at the vertical asymptote $x=-3$ .

  $\begin{array}{c}\underset{x\to -3^{-}}{\lim }f\left(x\right)=\underset{x\to -3^{-}}{\lim }\left(2-7g\left(x+3\right)\right)\\ =2-7\underset{x\to -3^{-}}{\lim }g\left(x+3\right)\\ \to 2-7\times -\infty \\ \to 2+\infty \\ \to \infty \end{array}$

  $\begin{array}{c}\underset{x\to -3^{+}}{\lim }f\left(x\right)=\underset{x\to -3^{+}}{\lim }\left(2-7g\left(x+3\right)\right)\\ =2-7\underset{x\to -3^{+}}{\lim }g\left(x+3\right)\\ \to 2-7\times \infty \\ \to 2-\infty \\ \to -\infty \end{array}$

Thus, $\underset{x\to -3^{-}}{\lim }f\left(x\right)\to \infty$ and $\underset{x\to -3^{+}}{\lim }f\left(x\right)\to -\infty$ .

Using limits, describe the behavior at the horizontal asymptote $y=2$ .

  $\begin{array}{c}\underset{x\to \pm \infty }{\lim }f\left(x\right)=\underset{x\to \pm \infty }{\lim }\left(2-7g\left(x+3\right)\right)\\ =2-7\underset{x\to \pm \infty }{\lim }g\left(x+3\right)\\ =2-7\times 0\\ =2\end{array}$

Thus, $\underset{x\to \pm \infty }{\lim }f\left(x\right)=2$ .

Graph:

  

Conclusion:

  $f\left(x\right)=2-7g\left(x+3\right)$ . The graph of $f$ is the graph of the reciprocal function shifted $3$ units to the left, then scaled vertically by a factor of $7$ , then reflected across the $x$ -axis, and finally translated $2$ units up.

The vertical asymptote is $x=-3$ and the horizontal asymptote is $y=2$ . $\underset{x\to -3^{-}}{\lim }f\left(x\right)\to \infty$ , $\underset{x\to -3^{+}}{\lim }f\left(x\right)\to -\infty$ , and $\underset{x\to \pm \infty }{\lim }f\left(x\right)=2$ describe the behavior at these asymptotes.

The graph of $f$ is shown above.