Determine whether the function is one-to-one. If it is, find a formula for its inverse.
\[
f(x)=x^{3}-1
\]
Is the function one-to-one?
Yes
No
Select the correct choice below and fill in any answer boxes within your choice.
A. The inverse function is $f^{-1}(x)=$
B. There is no inverse function.

Step 1 ：A function is one-to-one if it passes the horizontal line test, which means that each y-value is paired with exactly one x-value. In other words, no two different x-values have the same y-value.

Step 2 ：The function \(f(x)=x^{3}-1\) is a cubic function, which is always one-to-one. Therefore, it has an inverse function.

Step 3 ：To find the inverse function, we can swap x and y in the equation and solve for y. So, we have \(x=y^{3}-1\).

Step 4 ：Solving this equation for y will give us the inverse function. The solution to the equation gives three roots due to the cubic nature of the equation. However, since we are dealing with real numbers, we only consider the real root.

Step 5 ：The real root is \((x + 1)^{1/3}\), which is the inverse function.

Step 6 ：Final Answer: The inverse function is \(f^{-1}(x) = \boxed{(x + 1)^{1/3}}\).