Answer:
[f^{-1}(n) = sqrt[3]{n/2} + 2
Explanation:
To find the inverse function of \(f(n) = 2(n-2)^3\), let's start by expressing \(n\) in terms of \(f(n)\) and then solving for \(n\).
\[f(n) = 2(n-2)^3\]
First, isolate the term with \(n\):
\[f(n)/2 = (n-2)^3\]
Now, take the cube root of both sides to solve for \(n-2\):
\[(n-2) = \sqrt[3]{f(n)/2}\]
Finally, solve for \(n\):
\[n = \sqrt[3]{f(n)/2} + 2\]
So, the inverse function of \(f(n) = 2(n-2)^3\) is:
\[f^{-1}(n) = \sqrt[3]{n/2} + 2\]