Question

To determine

(a) To sketch the function \(y = \ln ( - x)\) using the given graphs; use the concept of transformations if needed.

(b) To sketch the function \(y = \ln |x|\) using the graphs from Figure 12 and Figure 13, use the concept of transformations if needed.

Short answer

(a) It is reflected about the \(y\)-axis.

(b) It is drawn on the same on the same screen.

Step-by-step solution

Given data

The given functions are \(y = \ln ( - x)\) and \(y = \ln |x|\).

Concept of Horizontal shift

Horizontal shift: When\(y = f(x + c)\)and\({\bf{c}} > {\bf{0}}\), shift the graph of\(y = f(x)\)a distance\(c\)units towards the left.

Plot of the graph of function \(y = \ln ( - x)\)

(a)

The graph of the function \(y = \ln x\) is shown below in Figure 1.

Then, draw the graph of \(y = \ln ( - x)\), by reflecting the graph \(y = \ln x\) about \(y\)-axis.

Thus, the graph of \(y = \ln ( - x)\) 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.

Plot of the graph of function \(y = \ln |x|\)

(b)

The absolute function \(y = \ln |x|\) is defined as, \(y = \left\{ {\begin{array}{*{20}{l}}{\ln x{\rm{ if }}x > 0}\\{\ln ( - x){\rm{ if }}x < 0}\end{array}} \right.\).

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

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

Therefore, the graph of \(y = \ln |x|\) is obtained by drawing the graphs of \(y = \ln x\) and \(y = \ln ( - x)\) 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.