or the function \( f(x)=\left(\frac{x}{5}\right)^{\frac{1}{6}} \), find \( f^{-1}(x) \) Answer \( \begin{array}{ll}f^{-1}(x)=5 x^{5} & f^{-1}(x)=(5 x)^{5} \\ f^{-1}(x)=\frac{x^{5}}{5} & f^{-1}(x)=5 x^{\frac{1}{5}}\end{array} \)

To find the inverse function \( f^{-1}(x) \) for \( f(x)=\left(\frac{x}{5}\right)^{\frac{1}{6}} \), we start with \( y = f(x) \), which gives us: \[ y = \left(\frac{x}{5}\right)^{\frac{1}{6}}. \] To isolate \( x \), we raise both sides to the power of 6: \[ y^6 = \frac{x}{5}. \] Then, we multiply both sides by 5: \[ x = 5y^6. \] Now, substituting \( y \) back with \( x \) to express the inverse function gives us: \[ f^{-1}(x) = 5x^6. \] So, the final answer is: \[ f^{-1}(x) = 5x^6. \] For the diverse interpretations you provided, none match the correct inverse function we've derived. Always check the order of operations when dealing with functions and their inverses to avoid common pitfalls in isolating variables!