$x=\frac{y+1}{y-2}$

$x(y-2)=y+1$

$xy-2x=y+1$

$xy-y=2x+1$

$y(x-1)=2x+1$

This function can be rewritten/simplified before finding the inverse.

$g(x)=\frac{5x(2-3x)}{5x^2}$