Chapter 1, Problem 16RE

To determine

To sketch: The graph of the function $f\left(x\right)=\{\begin{array}{l}-x\text{ if }x<0\\ e^x-1\text{ if }x\ge 0\end{array}$ by using transformation.

Explanation of Solution

Since the given function is the step function, there are two functions such as, $y=-x,\text{if }x<0$ and $y=e^x-1,\text{if }x\ge 0$.

The standard graph of the function $y=e^x$ and $y=x$ is roughly drawn and shown below in Figure 1.

Then, to draw the graph of $y=e^x-1,\text{if }x\ge 0$, shift the graph $y=e^x$ one unit downward and consider only the first quadrant portion .

Similarly, to draw the of $y=-x,\text{if }x<0$, reflect the graph $y=x$ about y-axis and consider the only the second quadrant part.

Thus, the graph of $f\left(x\right)=\{\begin{array}{l}-x\text{ if }x<0\\ e^x-1\text{ if }x\ge 0\end{array}$ is shown below in Figure 2.

From figure 2, it is observed that the graph $y=e^x-1$ is drawn for the numbers $x\ge 0$ and the graph $y=-x$ is drawn for the numbers $x<0$.