Chapter 6.3, Problem 34E

To determine

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

Answer to Problem 34E

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 $y<-3$ .

Explanation of Solution

Giveninformation:

The function is $f(x)=-{\left(\frac{1}{2}\right)}^{x+1}-3$ .

To graph the function, let’s begin by graphing $y={\left(\frac{1}{2}\right)}^x$ . For this, we prepare the table values as follows:

$x$	$-2$	$-1$	0	1	2
$f(x)$	4	2	1	$\frac{1}{2}$	$\frac{1}{4}$

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

  

Now, shift the graph by 2 units to the right and flip it, then, shift it 3 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 $y=-3$ but not intersect it. Therefore, the domain is all real numbers and the range is $y<-3$ .