as you should already know, to get the inverse relation of any expression, we start off by doing a quick switcheroo on the variables, and then solve for "y".
[tex]\stackrel{f(x)}{y}=2+\sqrt{x-1}~~\implies ~~\stackrel{\textit{quick switcheroo}}{x = 2+\sqrt{y-1}}\implies x-2=\sqrt{y-1} \\\\\\ \stackrel{\textit{squaring both sides}}{(x-2)^2=y-1}\implies (x-2)^2+1=y\implies (x-2)(x-2)+1=y \\\\\\ \stackrel{F~O~I~L}{(x^2-4x+4)}+1=y\implies x^2-4x+5=\stackrel{y~\hfill }{f^{-1}(x)}[/tex]
