To factor this, start by replacing x with y and then solve for y
[tex]f(x) = \frac{2x+1}{x - 1}[/tex]
[tex]x = \frac{2y+1}{y - 1}[/tex]
[tex](y-1)x = \frac{2y+1}{y - 1} * (y-1)[/tex]
[tex](y-1)x = 2y+1[/tex]
[tex]xy -x = 2y+1[/tex]
[tex]xy - 2y -x +x = 2y - 2y +x+1[/tex]
[tex]xy - 2y = x+1[/tex]
Factor out y on the left side.
[tex]\frac{xy - 2y}{y} = x+1[/tex]
[tex]y(x - 2) = x+1[/tex]
[tex]\frac{y(x - 2)}{(x-2)} = \frac{x+1}{x-2}[/tex]
[tex]y = \frac{x+1}{x-2}[/tex]
[tex]f(x)^{-1} = \frac{x+1}{x-2}[/tex]
