Chapter 1.6, Problem 30E

To determine

To sketch: The graph of $f^{-1}$ from the given graph $f$.

Explanation of Solution

Identify the points from the given graph.

The points plotted in the graph are (0, 1), (2, 0) and (4, −1).

Interchange the x and y values to find the points of inverse graph.

Thus, the inverse function has, (1, 0), (0, 2) and (−1, 4).

Also notice that the domain of the graph $f$ is [0, 4] and the range of the graph $f$ is [−1, 1].

Thus, the domain of the graph $f^{-1}$ is [−1, 1] and the range of the graph $f^{-1}$ is [0, 4].

So, to sketch the graph of $f^{-1}$, draw the line y = x and plot the points (1, 0), (0, 2) and (−1, 4).

Thus, the graph of $f^{-1}$ is shown below in Figure 1.

From Figure 1, notice that the graph of $f^{-1}$ is the reflection of the graph of f about y = x. Also, notice that the graph f follows the cosine curve.