Question

In Exercises \(35-40,\) 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. $$ f(x)=\sqrt[3]{x-1} $$

Short answer

The inverse of the function \(f(x) = \sqrt[3]{x-1}\) is \(f^{-1}(x) = x^3 + 1\). When graphed, these functions are reflections of each other in the line \(y = x\).

Step-by-step solution

Finding the Inverse of the Function

To find the inverse of the function \(f(x) = \sqrt[3]{x - 1}\), we first replace \(f(x)\) with \(y\), this gives us \(y = \sqrt[3]{x - 1}\). Then interchange the roles of \(x\) and \(y\) which leads to \(x = \sqrt[3]{y - 1}\). Now, solve this for \(y\) to get the inverse.

Solving for the Inverse

To isolate \(y\), you will cubic both sides. So, \(x^3 = y - 1\). Then isolate \(y\) by adding 1 to both sides of the equation, which gives us \(y = x^3 + 1\). This is the inverse of the original function, \(f^{-1}(x) = x^3 + 1\).

Graphing the functions

To satisfy the second part of the exercise, both the original function, \(f(x) = \sqrt[3]{x-1}\), and its inverse, \(f^{-1}(x) = x^3 + 1\), need to be graphed on the same viewing window on a graphing utility. Note that the original and its inverse function will be reflections of each other in the line \(y = x\). This is because, in an inverse function, the roles of \(x\) and \(y\) are switched.