Question

(a) Graph the function\(f\left( x \right) = {e^x} + {\bf{ln}}\left| {x - {\bf{4}}} \right|\)for \({\bf{0}} \le x \le {\bf{5}}\). Do you think the graph is an accurate representation of f?

(b) How would you get a graph that represents f better?

Short answer

(a) The graph is shown below:

The graph does not represent the discontinuity at \(x = 4\), and it can be represented better in a rectangle of range \(3.5 < x < 4.1\).

(b) For large negative values, the graph follows the path of \(\ln \left| {x - 4} \right|\) whereas for \(x > 5\) it shows the graph of \({e^x}\). The discontinuity in the graph at \(x = 4\) can be better represented as shown below:

Step-by-step solution

Sketch the graph of \(f\left( x \right) = {e^x} + {\bf{ln}}\left| {x - {\bf{4}}} \right|\)

Use the following steps to plot the graph of given functions:

1. In the graphing calculator, select “STAT PLOT” and enter the equation\({e^X} + \ln \left| {X - 4} \right|\) in the \({Y_1}\) tab.
2. Enter the graph button in the graphing calculator.

The figure below represents the graph of f.

The first graph is the calculator-produced graph, but the function has an infinite discontinuity at \(x = 4\).

The second graph represents the graph of f using the increasing points, showing that the graph has a discontinuity\(x = 4\).

 Step 2: Sketch the graph to find the discontinuity

In order to show all features of \(f\left( x \right)\) several graphs are required.

It can be observed that for large negative values, the graph follows the curve of \(\ln \left| {x - 4} \right|\), whereas for \(x > 5\) the graph follows \({e^x}\).

In order to represent the discontinuity at \(x = 4\)a small rectangle,a view is required.

Thus, the graph is obtained.