Question

To sketch the graph of \({f^{ - 1}}\) from the given graph of \(f\).

Short answer

The graph of \({f^{ - 1}}\) is shown below in Figure 1.

Step-by-step solution

Given function

The given function is \({f^{ - 1}}\).

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.

Definition of Graph

Agraph can be defined as a pictorial representation or a diagram that represents data or values in an organized manner.

The points on the graph often represent the relationship between two or more things.

Plot the graph from the given values

Identify the points from the given graph.

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

Interchange\(x\)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.