Chapter 7.5, Problem 11PPS

To determine

To graph the function $f(x)=2\text{ · }{\left(\frac{1}{6}\right)}^x$ , find its domain, range and equation of the asymptote.

Explanation of Solution

Given:

Function: $f(x)=2\text{ · }{\left(\frac{1}{6}\right)}^x$

Calculation for graph:

Consider $y=2\text{ · }{\left(\frac{1}{6}\right)}^x$

Values of x	Values of y
0	2
1	0.33
2	0.055
3	0.00925
-1	12

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

Graph:

  

Interpretation:

Domain:

The domain of the expression $f(x)=2\text{ · }{\left(\frac{1}{6}\right)}^x$ 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: $(0,\infty )$ for all y, $y>0$

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 $y=0.$