Find inverse functions by solving f(x)=c for simple invertible functions and writing inverse expressions.
Find the inverse of a square-root function by swapping x and y and solving
Problem
Find the inverse of \(f(x)=\sqrt{x}+2\). State the original domain and range, write \(y=f(x)\), swap \(x\) and \(y\), isolate and square the radical equation, then give \(f^{-1}\) with its domain and range and verify \(f^{-1}(f(9))\).
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What this problem is really about
Inverting a square-root function turns its one-sided range into a restriction on the resulting quadratic. We’ll record the original domain and range, swap the variables, isolate the radical before squaring, and use the exchanged restrictions to keep the inverse on the correct branch.
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