Answer
$f^{-1}(x)=2^{x}+1$
Work Step by Step
To find the inverse of y=f(x):
1. swap x and y in the expression of the function
2. solve for y
3. replace $y $with $f^{-1}(x)$
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$y=\log_{2}(x-1)$
$ x=\log_{2}(y-1)\qquad$... apply $2^{(..)}$ to both sides
$ 2^{x}=y-1\qquad$... $/+1$
$y=2^{x}+1$
$f^{-1}(x)=2^{x}+1$