You must check if a function is injective as well surjective before finding its inverse. But here they have already asked to find inverse so we are not worried about its bijectivity.
let,
[tex]y= \frac{x-2}{3x}[/tex]
[tex]3xy= x-2[/tex]
[tex]x(3y-1) = -2[/tex]
[tex]x= \frac{-2}{3y-1}[/tex]
Now replace x with y and rewrite.
[tex]y = \frac{-2}{3x-1} [/tex]
Therefore, inverse of g(x) is,
g⁻¹(x) = [tex] \frac{-2}{3x-1} [/tex]
