Chapter 7.5, Problem 17PPS

To determine

To graph the function $y=\frac{1}{2}(2^x-8)$ , find its domain, range and equation of the asymptote.

Explanation of Solution

Given:

Function: $y=\frac{1}{2}(2^x-8)$

Calculation for graph:

Consider $y=\frac{1}{2}(2^x-8)$

Values of x	Values of y
0	-3.5
1	-3
2	-2
-1	-3.75

By taking different values of x , the graph can be plotted.

Graph:

  

Interpretation:

Domain:

The domain of the expression $y=\frac{1}{2}(2^x-8)$ is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Hence,

Domain: $(-\infty ,\infty )$

Range is the set of all y -values that the function passes through.

Hence,

Range: $(-4,\infty )$ for all y, $y>-4$

Asymptotes:

Exponential functions have only horizontal asymptotes.

So, there will be no vertical asymptotes, since the function do not have a denominator.

Therefore, equation of horizontal asymptote is $y=0.$