Chapter 1.6, Problem 48E

Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3.

48. (a) y = ln(–x)

(b) y = ln |x|

To determine

To sketch: The function $y=\ln \left(-x\right)$ using the graphs from Figure 12 and Figure 13; use the concept of transformations if needed.

Explanation of Solution

From Figure 13, the graph of the function $y=\ln x$ is shown below in Figure 1.

Then, draw the graph of $y=\ln \left(-x\right)$, by reflecting the graph $y=\ln x$ about y-axis. Thus, the graph of $y=\ln \left(-x\right)$ is shown below in Figure 2.

Observe that Figure 2 is obtained from Figure 1 in such a way that it is reflected about the y-axis.

To determine

To sketch: The function $y=\ln \left|x\right|$ using the graphs from Figure 12 and Figure 13; use the concept of transformations if needed.

Explanation of Solution

The absolute function $y=\ln \left|x\right|$ is defined as, $y=\{\begin{array}{l}\ln x\text{if }x>0\\ \ln \left(-x\right)\text{if }x<0\end{array}$.

From part (a), identify that the graph of $y=\ln x$ exist when $x>0$ and the graph $y=\ln \left(-x\right)$ exist when $x<0$.

Thus, the graph $y=\ln \left|x\right|$ is a combination of the graphs $y=\ln x$ and $y=\ln \left(-x\right)$.

Therefore, the graph of $y=\ln \left|x\right|$ is obtained by drawing the graphs of $y=\ln x$ and $y=\ln \left(-x\right)$ on the same screen as shown below in Figure 3.

Observe that Figure 3 is obtained from Figure 1 and Figure 2 in such a way that it is drawn on the same screen.