Determine whether the function is one-to-one. If so, (a) write an equation for the inverse function in the form $y=f^{-1}(x)$ and (b) graph $f$ and $f^{-1}$ on the same axes. If the function is not one-to-one, say so.
\[
f(x)=x^{3}-6
\]
(a) Write an equation for the inverse function in the form $y=f^{-1}(x)$. Select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.
Answer
Final Answer: The inverse function is \(f^{-1}(x) = \boxed{(x+6)^{1/3}}\).
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 \(f^{-1}(x) = \boxed{(x+6)^{1/3}}\).
