Find inverse functions for simple invertible functions, including rational examples.
Find the inverse of a restricted quadratic by swapping x and y
Problem
Find \(f^{-1}(x)\) and its domain for \(f(x)~=~(x~-~2)^2~+~3\) restricted to \(x~\ge~2\).
Big Picture
What this problem is really about
A quadratic becomes invertible only after its domain restriction keeps one branch. Record the original range before swapping variables, because that range becomes the inverse domain. When solving the swapped equation, the same branch restriction determines which square-root sign is valid; keeping both signs would create a relation instead of a function.
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