Problem 48 - 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 of the function \(f(x)=(x-2)^3\) is \(f^{-1}(x)=\sqrt[3]{x}+2\). The domain and range of both the function and its inverse are \(-\infty,+\infty\) in interval notation.

Step 1: Find the inverse of the function

To find the inverse of the function \(f(x)=(x-2)^3\), replace \(f(x)\) with \(y\) to get \(y=(x-2)^3\). Then swap \(x\) and \(y\) to find the inverse, i.e., \(x=(y-2)^3\). Solve this equation for \(y\) to obtain the inverse function. Taking the cube root of both sides yields \(y=\sqrt[3]{x}+2\). Thus, \(f^{-1}(x)=\sqrt[3]{x}+2\).

Step 2: Graph the function and its inverse

The graph of the function \(f(x)=(x-2)^3\) is a cubic curve shifted 2 units to the right. The graph of the inverse function \(f^{-1}(x)=\sqrt[3]{x}+2\) is similarly a cubic curve, but with a horizontal stretch and shifted 2 units upward. They are reflections of each other over the line \(y=x\)

Step 3: Determine the domain and range

The domain of the function \(f(x)=(x-2)^3\) is all real numbers, as there are no restrictions on the values \(x\) can take. The range is also all real numbers, as cubing any real number can also produce any real number. Therefore, the domain and range of \(f(x)\) are both \(-\infty,+\infty\). For the inverse function \(f^{-1}(x)=\sqrt[3]{x}+2\), the domain and range are also \(-\infty,+\infty\) because cube root function takes real inputs and gives real outputs.