For the function \( f(x)=\frac{\sqrt[5]{x}}{9} \), find \( f^{-1}(x) \) Answer \[ \begin{array}{ll}f^{-1}(x)=\left(\frac{x}{9}\right)^{5} & f^{-1}(x)=(9 x)^{5} \\ f^{-1}(x)=\frac{x^{5}}{9} & f^{-1}(x)=9 x^{5}\end{array} \]

To find the inverse function \( f^{-1}(x) \) for \( f(x) = \frac{\sqrt[5]{x}}{9} \), we start by letting \( y = f(x) \). Thus, we write: \[ y = \frac{\sqrt[5]{x}}{9}. \] To isolate \( x \), we first multiply both sides by 9: \[ 9y = \sqrt[5]{x}. \] Next, we raise both sides to the fifth power to eliminate the root: \[ (9y)^5 = x. \] Thus, we obtain the inverse function: \[ f^{-1}(x) = (9x)^5. \] As for your additional information: Did you know that the concept of inverse functions can be traced back to ancient mathematics? The Greeks, like Euclid, explored the properties of numbers and geometric shapes, laying the groundwork for function theory that we build on today! And if you ever struggle with calculating inverse functions, remember this simple tip: Always swap the roles of \( x \) and \( y \) in the equation, solve for \( y \), and you'll often find yourself on the right path to uncovering the hidden inverse!