Question

Find the inverse function of \(f\). Use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window. Describe the relationship between the graphs. $$ f(x)=x^{2 / 3}, \quad x \geq 0 $$

Short answer

The inverse function of \(f(x) = x^{2 / 3}\) is \(f^{-1}(x) = x^{3 / 2}\). The graph of \(f^{-1}\) is a reflection of the graph of \(f\) across the line \(y = x\).

Step-by-step solution

Calculation of Inverse Function

Let's replace \(f(x)\) with \(y\) giving us \(y = x^{2/3}\). Next, swap \(x\) and \(y\) which will give us \(x = y^{2/3}\). Solving for \(y\), we have \(y = x^{3/2}\). Thus, the inverse function \(f^{-1}(x) = x^{3 / 2}\).

Graph the original and inverse function.

First draw the graph of the original function which is \(y = x^{2 / 3}\). This will be a curve starting from (0,0), going upwards and to the right. Then plot the inverse function which is \(y = x^{3 / 2}\). This will also be a curve starting from (0,0), but it will steeply rise upwards and to the right.

Describe the relationship between the graphs

Upon comparison, it can be observed that the graph of the inverse function is a reflection of the graph of the original function across the line \(y = x\). They intersect at (0,0), and for any point (a, b) on the graph of the original function, there will be a point (b, a) on the graph of the inverse function.