Problem 46 - College Algebra Essentials

Question

a. Find an equation for \(f^{-1}(x)\) b. Graph \(f\) and \(f^{-1}\) in the same rectangular coordinate system. c. Use interval notation to give the domain and the range of \(f\) and \(f^{-1}\) $$f(x)=x^{3}+1$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x) = \sqrt[3]{x-1}\). The domains of both \(f(x)\) and \(f^{-1}(x)\) are all real numbers, as are their ranges. You would see the reflection of the graph of \(f\) over the line \(y=x\) when you graph \(f^{-1}\).

Step 1: Find the Inverse Function

To find the inverse function, replace \(f(x)\) with \(y\) to get \(y=x^{3}+1\). Then, switch the roles of \(x\) and \(y\) to obtain \(x=y^{3}+1\). Now solve for \(y\) to find \(f^{-1}(x)\). Subtract 1 from both sides to get \(x-1=y^{3}\). Finally, take the cubed root of both sides to find \(f^{-1}(x) = \sqrt[3]{x-1}\).

Step 2: Graph the Functions

Plot the function \(f(x)=x^{3}+1\) and its inverse \(f^{-1}(x) = \sqrt[3]{x-1}\) on the same graph. Remember that the graph of an inverse function is a reflection of the original function over the line \(y=x\). This is because for every point (a, b) on the graph of the original function, there will be a point (b, a) on the graph of the inverse function.

Step 3: Determine the Domains and Ranges

The domain of a function is the set of all possible x-values. For the function \(f(x)=x^{3}+1\), this is all real numbers, or \(-\infty \leq x \leq \infty\). The range is the set of all possible y-values. Again, for the function \(f(x)=x^{3}+1\), this is all real numbers. For the inverse function \(f^{-1}(x) = \sqrt[3]{x-1}\), the domain and range also include all real numbers, as you can cube root any real number, and subtracting 1 does not limit the possible values.