Problem
Question 4 of 11, Step 1 of 1 Correct Find a formula for the inverse of the following function, if possible. \[ V(x)=\left(x^{3}-3\right)^{\frac{1}{3}}-1 \] Answer How to enter your answer (opens in new window) Keypad Keyboard Shortcuts Selecting a radio button will replace the entered answer value(s) with the radio button value. If the radio button is not selected, the entered answer is used. \[ V^{-1}(x)= \] does not have an inve
Solution
Step 1 ：Replace \(V(x)\) with \(y\) to get \(y = (x^{3} - 3)^{\frac{1}{3}} - 1\).
Step 2 ：Swap \(x\) and \(y\) to get \(x = (y^{3} - 3)^{\frac{1}{3}} - 1\).
Step 3 ：Solve for \(y\) to get the inverse function. The equation becomes \(x = ((y^{3} - 3)^{\frac{1}{3}} - 1)^{3} - 3)^{\frac{1}{3}} - 1\).
Step 4 ：Solving this equation gives three roots, but we are interested in the real root. The real root is the first element in the list of roots.
Step 5 ：The inverse of the function is \(V^{-1}(x)=1.44224957030741*(0.333333333333333*(x + 1.0)^{3} + 1)^{\frac{1}{3}}\).
Step 6 ：\(\boxed{V^{-1}(x)=1.44224957030741*(0.333333333333333*(x + 1.0)^{3} + 1)^{\frac{1}{3}}}\) is the final answer.
