Question 4 of 11, Step 1 of 1
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Find a formula for the inverse of the following function, if possible.
$V (x) = {(x^{3} - 3)}^{\frac{1}{3}} - 1$

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$V^{- 1} (x) =$
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Answer

$V^{- 1} (x) = 1.44224957030741 * (0.333333333333333 * (x + 1.0)^{3} + 1)^{\frac{1}{3}}$ is the final answer.

Steps

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 ： $V^{- 1} (x) = 1.44224957030741 * (0.333333333333333 * (x + 1.0)^{3} + 1)^{\frac{1}{3}}$ is the final answer.