Chapter 3.1, Problem 29E

Solution

a.

To match: the given function with its graph.

Graph (b)

Given information:

The given function is $y=3^{-x}-2$ .

The given graphs are:

1. 
2. 
3. 
4. 
5. 
6. 

Explanation:

Graph the given function using Ti-83 calculator,

First pressand enter the function as shown:

  

Then press and set the window as shown:

  

Lastly press and this gives the required graph.

  

This graph matches with the graph (b).

b.

To match: the given function with its graph without using grapher.

Graph (b)

Given information:

The given function is $y=3^{-x}-2$ .

The given graphs are:

1. 
2. 
3. 
4. 
5. 
6. 

Explanation:

Consider the given function,

  $y=3^{-x}-2$

Here, it can be observed that at $x=0$, the value of this function is:

  $\begin{array}{c}y=3^{-0}-2\\ =1-2\\ =-1\end{array}$

The only graphs for which this is not true are graphs (a) and (d). So, these choices are eliminated.

Now, check what happens to function as x becomes extremely large positive number and as x becomes extremely large negative number

  $\begin{array}{c}\underset{x\to \infty }{\lim }\left(3^{-x}-2\right)=\underset{x\to \infty }{\lim }\left(3^{-x}\right)-2\\ =0-2\\ =-2\end{array}$

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

The only graph with this behavior is graph (b).