Chapter 4.3, Problem 58E

To determine

To find: The graph for the function and the find he domain, range, and asymptote.

Answer to Problem 58E

The required graph is shown in Figure 2, the domain of the function is $\left(1,\infty \right)$ and its range is $\left(-\infty ,-\infty \right)$ . The vertical asymptotes of the graph is $x=1$ .

Explanation of Solution

Given:

The given function is $y={\log }_3\left(x-1\right)-2$ .

Calculation:

Consider the given function is,

  $y={\log }_3\left(x-1\right)-2$

The parent function for the given function is $y={\log }_{3}x$ . The graph for $y={\log }_3\left(x-1\right)-2$ is obtained by shifting the function $y={\log }_{3}x$ to the right of the unit function.

The graph for the $y={\log }_{3}x$ is shown in Figure 1

Figure 1

The graph for the function $y={\log }_3\left(x-1\right)-2$ is shown in Figure 2

Figure 2

From the above graph, the vertical asymptote of the function $y={\log }_3\left(x-1\right)-2$ is,

  $x=1$

The input of the $x$ extends from 1 to 8 where 1 and 8 are not excluded. Thus, the domain of the function is,

  $\left(1,\infty \right)$

The range of the given function $y={\log }_3\left(x-1\right)-2$ is,

  $\left(-\infty ,\infty \right)$