Chapter 4.3, Problem 61E

To determine

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

Answer to Problem 61E

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

Explanation of Solution

Given:

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

Calculation:

Consider the parent function for given function is,

  $y=\left|\ln x\right|$

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

  $f\left(x\right)=\ln x$

The graph for the parent function is shown in Figure 1

Figure 1

The function $y=\ln x$ in terms of function is $f\left(x\right)=\ln x$ is,

  $\begin{array}{c}y=\left|\ln \left(x\right)\right|\\ =\left|f\left(x\right)\right|\left(\mathrm{s}\text{ince,}f\left(x\right)=\ln \left(x\right)\right)\end{array}$

The graph for the function $y=\left|\ln x\right|$ is similar to the graph of $y=\ln x$ with the negative part of the graph is reflected across the $x$ axis as shown in Figure 2

Figure 2

From the above graph, the asymptote of the function $y=\left|\ln \left(x\right)\right|$ is,

  $x=0$

The domain of the function is,

  $\left(0,\infty \right)$

The range of the given function $y=\left|\ln \left(x\right)\right|$ is,

  $\left(0,\infty \right)$