Chapter 4.3, Problem 62E

To determine

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

Answer to Problem 62E

The required graph is shown in Figure 2, the domain of the function is $\left(-\infty ,0\right)\cup \left(0,\infty \right)$ and its range is $\left(-\infty ,\infty \right)$ . The vertical asymptotes of the graph is $x=0$ .

Explanation of Solution

Given:

The given function is $y=\ln \left|x\right|$ .

Calculation:

Consider the parent function for given function is,

  $y=\ln \left|x\right|$

The parent function for the given function $y=\ln \left|x\right|$ is,

  $h\left(x\right)=\ln x$

The graph for the parent function is shown in Figure 1

Figure 1

For the function $g\left(x\right)=\left|x\right|$ , the function is symmetric about the $y$ axis. The graph for it is shown n Figure 2

Figure 2

The function $f\left(x\right)=\ln \left|x\right|$ in terms of function is $g\left(x\right)=\left|x\right|$ and $f\left(x\right)=\ln x$ that is,

  $h\left(x\right)=f\left(g\left(x\right)\right)$

The graph for the function $h\left(x\right)$ is,

Figure 3

From the figure the set of the functions when the graphs defined for $x<0$ or $x>0$ . Then, the domain for the graph $g\left(x\right)=\ln \left|x\right|$ is,

  $\left(-\infty ,0\right)\cup \left(0,\infty \right)$

The range of the graph is,

  $\left(-\infty ,\infty \right)$

From the above graph the asymptote of the graph is,

  $x=0$