Chapter 6.3, Problem 35E

To determine

To sketch:The graph of the function and describe the domain and range.

Answer to Problem 35E

The graph of the given function is drawn in blue color in the following graph:

  

The domain is all real numbers and the range is $y<-2$ .

Explanation of Solution

Giveninformation:

The function is $f(x)=-8{\left(0.75\right)}^{x+2}-2$ .

To graph the function, let’s begin by graphing $y=-8{\left(0.75\right)}^x$ . For this, we prepare the table values as follows:

$x$	$-2$	$-1$	0	1	2
$f(x)$	$-14.22$	$-10.67$	$-8$	$-6$	$-4.5$

Now, plotting the ordered points and drawing a smooth curve through the points, we obtain:

  

Now, shift the graph by 2 units to the left and shift it 2 units down to obtain the graph of the given function. That is,

  

Conclusion:

From the graph, we can observe that the graph of the function approaches the line $y=-2$ but not intersect it. Therefore, the domain is all real numbers and the range is $y<-2$ .