Step 1 :To find the inverse of a function, we need to switch the roles of x and y (or f(x)) and solve for y. In this case, we need to solve the equation \(x = 9 + y^{1/3}\) for y.
Step 2 :Solving the equation gives us the inverse function as \(f^{-1}(x) = 729.0*(0.111111111111111*x - 1)^3\).
Step 3 :\(\boxed{f^{-1}(x) = 729.0*(0.111111111111111*x - 1)^3}\) is the inverse function of \(f(x)=9+\sqrt[3]{x}\).