Problem 47 - 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 \(f^{-1}\) $$f(x)=(x+2)^{3}$$

Step-by-Step Solution

Answer

The inverse function is \(f^{-1}(x)= \sqrt[3]{x} -2\). The domain and range for both \(f(x)\) and \(f^{-1}(x)\) are all real numbers.

Step 1: Find the Inverse Function

The given function is \(f(x)=(x+2)^{3}\). Swap \(x\) and \(y\) to find \(f^{-1}(x): x=(y+2)^{3} \). Solve this for \(y\) to get the inverse function \(f^{-1}(x)=y=\sqrt[3]{x}-2\).

Step 2: Sketch the Graphs

To graph \(f(x)\) and \(f^{-1}(x)\), a set of points need to be plotted for both the functions. For \(f(x)\), you can use points like \((-3, -1)\), \((-2, 0)\), \((-1, 1)\), \((0, 8)\). Similarly for \(f^{-1}(x)\), you can use points like \((-1, -3)\), \((0, -2)\), \((1, -1)\), \((8, 0)\). Note that the graph of \(f^{-1}\) is a reflection of graph \(f\) about the line \(y=x\).

Step 3: Find the Domain and Range

For the function \(f(x)\), the domain is all real numbers because a cubic function is defined for all real values of \(x\). Accordingly, the range is all real numbers. For the inverse function \(f^{-1}(x)\), the domain and range swap places. So the domain is all real numbers and the range is also all real numbers.