Step 1 :The function \(f(x)=x^{3}-6\) is a one-to-one function because the cubic function is always one-to-one.
Step 2 :The inverse of a function is found by swapping the x and y values, so to find the inverse of this function, we need to solve the equation \(y=x^{3}-6\) for x.
Step 3 :The solution gives three roots due to the cubic equation. However, considering the real root, the inverse function is \(f^{-1}(x) = (x+6)^{1/3}\).
Step 4 :Final Answer: The inverse function is \(f^{-1}(x) = \boxed{(x+6)^{1/3}}\).