Answer
Yes, $f$ and $g$ are inverse functions by the Inverse Function Property.
Work Step by Step
We know that if $f$ and $g$ are inverse functions of each other, then $f(g(x))=g(f(x))=x$.
$\displaystyle f(g(x))=\frac{1}{(\frac{1}{x}+2)-2}=\frac{1}{\frac{1}{x}}=x$
$\displaystyle g(f(x))=\frac{1}{\frac{1}{x-2}}+2=x-2+2=x$
Therefore, $f$ and $g$ are inverse functions.
