Chapter 2.2, Problem 55E

To determine

To graph: The function $y=f\left(x\right)$ such that $\underset{x\to 1}{\lim }f\left(x\right)=2$ , $\underset{x\to 5^{-}}{\lim }f\left(x\right)=\infty$ , $\underset{x\to 5^{+}}{\lim }f\left(x\right)=\infty$ , $\underset{x\to \infty }{\lim }f\left(x\right)=-1$ , $\underset{x\to -2^{+}}{\lim }f\left(x\right)=-\infty$ , $\underset{x\to -2^{+}}{\lim }f\left(x\right)=\infty$ and $\underset{x\to -\infty }{\lim }f\left(x\right)=0$ .

Explanation of Solution

Given information:

The given conditions are $\underset{x\to 1}{\lim }f\left(x\right)=2$ , $\underset{x\to 5^{-}}{\lim }f\left(x\right)=\infty$ , $\underset{x\to 5^{+}}{\lim }f\left(x\right)=\infty$ , $\underset{x\to \infty }{\lim }f\left(x\right)=-1$ , $\underset{x\to -2^{+}}{\lim }f\left(x\right)=-\infty$ , $\underset{x\to -2^{+}}{\lim }f\left(x\right)=\infty$ and $\underset{x\to -\infty }{\lim }f\left(x\right)=0$ .

Graph:

Assume that the function is $f\left(x\right)=\{\begin{array}{l}-\frac{1}{x+2},x<-2\\ \frac{1}{x-5}-1,x>5\\ \frac{x^5}{\left(x+2\right)\left(x-5\right)},-2<x<5\end{array}$ that satisfies all the given conditions.

To graph a function $f\left(x\right)=\{\begin{array}{l}-\frac{1}{x+2},x<-2\\ \frac{1}{x-5}-1,x>5\\ \frac{x^5}{\left(x+2\right)\left(x-5\right)},-2<x<5\end{array}$ , follow the steps using graphing calculator.

First press “ON” button on graphical calculator and press $\boxed{\text{Y}=}$ key.

Enter the keystrokes given below after the symbol ${\text{Y}}_1$ .

  $\boxed{(}\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{+}\boxed{2}\boxed{)}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{5}\boxed{(-)}\boxed{2}\boxed{)}$

Enter the keystrokes given below after the symbol ${\text{Y}}_2$ .

  $\boxed{(}\boxed{(-)}\boxed{1}\boxed{+}\boxed{(}\boxed{1}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{5}\boxed{)}\boxed{)}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{3}\boxed{5}\boxed{)}$

Enter the keystrokes given below after the symbol ${\text{Y}}_3$ .

  $\begin{array}{l}\boxed{(}\boxed{\text{X}}\boxed{^}\boxed{5}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{+}\boxed{2}\boxed{)}\boxed{(}\boxed{\text{X}}\boxed{-}\boxed{5}\boxed{)}\boxed{)}\boxed{÷}\boxed{(}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{3}\boxed{(-)}\boxed{2}\boxed{\text{X}}\\ \boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{\to }\boxed{1}\boxed{\text{X}}\boxed{2\text{nd}}\boxed{\text{MATH}}\boxed{5}\boxed{5}\boxed{)}\end{array}$

The display will show the equations,

  $\begin{array}{l}{\text{Y}}_{\text{1}}=\left(1/\left(\text{X}+2\right)\right)/\left(\text{X}<-2\right)\\ {\text{Y}}_2=\left(-1+\left(1/\left(\text{X}-5\right)\right)\right)/\left(\text{X}>5\right)\\ {\text{Y}}_3=\left(\text{X}^5/\left(\text{X}+2\right)\left(\text{X}-5\right)\right)/\left(\text{X}>-2\text{and X}<5\right)\end{array}$

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph of given function in standard window as shown in Figure (1).

  

Figure(1)

Interpretation: From the above graph it can be observed that the function that satisfies all the given conditions is a piecewise function.