Find the inverse function of \( f(x)=9+\sqrt[3]{x} \). \( f^{-1}(x)=\square \) Basic Funcs Trig

To find the inverse function of \( f(x) = 9 + \sqrt[3]{x} \), start by replacing \( f(x) \) with \( y \): \[ y = 9 + \sqrt[3]{x} \] Next, solve for \( x \) in terms of \( y \): \[ y - 9 = \sqrt[3]{x} \] Now, cube both sides to eliminate the cube root: \[ (y - 9)^3 = x \] Finally, swap \( x \) and \( y \) to express the inverse function: \[ f^{-1}(x) = (x - 9)^3 \] So, the inverse function is: \( f^{-1}(x) = (x - 9)^3 \) Now you have \( f^{-1}(x) \) neatly boxed and ready to go! Understanding inverses is like flipping a switch: just as functions transform input to output, inverses flip back from output to input. It’s essential in many areas, such as solving equations, where you might need to "undo" a function to find original values or inputs. One common mistake when finding inverses is forgetting to switch the roles of \( x \) and \( y \) at the end. Always check that you’ve properly re-expressed the function and retained the operations so that the inverse correctly undoes the original function’s transformations!