Answer
Using inverse function we have:
$f^{-1}(x)=\frac{2x+2}{x-1}$
$g^{-1}(x)=\frac{x+2}{x-2}$
So these functions are inverse of each other.
Work Step by Step
To show it we will simply calculate inverse of the both $f$ and $g$ functions.
To calculate $f^{-1}(x)$ first we have to write it in terms of $y$ and $x$ and then replace $x$ with $y$ and vice versa.
$f(x)=\frac{x+2}{x-2}$
$y=\frac{x+2}{x-2}$
$x=\frac{y+2}{y-2}$
$x(y-2)=y+2$
$xy-2x=y+2$
$xy-y=2x+2$
$y(x-1)=2x+2$
$y=\frac{2x+2}{x-1}$
$f^{-1}(x)=\frac{2x+2}{x-1}$
We will follow the same way to calculate $g^{-1}(x)$
$g(x)=\frac{2x+2}{x-1}$
$y=\frac{2x+2}{x-1}$
$x=\frac{2y+2}{y-1}$
$x(y-1)=2y+2$
$xy-x=2y+2$
$xy-2y=2+x$
$y(x-2)=x+2$
$y=\frac{x+2}{x-2}$
$g^{-1}(x)=\frac{x+2}{x-2}$
