Remark
The reason you are not getting many replies to this is an uncertainty about f(x). We do not know if you mean f(x) = 2/(x - 6) or f(x) = (2/x) - 6. We'll try it both ways.
First Way f(x) = 2/(x - 6)
f(x = y = 2/(x - 6) Interchange x and y
x = 2 / (y - 6) Multiply both sides by y - 6
x(y - 6) = 2 Divide by x
y - 6 = 2/x Add 6 to both sides.
y = 2/x + 6 Which is close to g(x) but it is not the same thing, even if you use x as a common denominator. That would give you
Second Way: f(x) = (2/x) - 6
f(x) = y = (2/x) - 6 Interchange x and y
x = (2/y) - 6 Add 6 to both sides
x + 6 = 2/y Multiply both sides by y
y(x + 6) = 2 Divide by sides by (x + 6)
y = 2/(x + 6)
Comment
These do not look anything alike so g(x) and f(x) are not inverses.