Chapter 7.5, Problem 2GP

To determine

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

Explanation of Solution

Given:

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

Calculation for graph:

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

Values of x	Values of y
0	0
1	-0.5
2	-0.75
3	-0.875
-1	1

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

Graph:

  

Interpretation:

Domain:

The domain of the expression $y={\left(\frac{1}{2}\right)}^x-1$ 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: $(-1,\infty )$ for all y, $y>-1$

Asymptotes:

Exponential functions have only horizontal asymptotes.

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

Therefore, the equation of horizontal asymptote is $y=-1.$