Inverse function of this function?

Don't worry that the answer is the same as the question itself. It just so happens that you are quite right :)

The inverse function of g(x) in your case is indeed g(x)!

g(x)=y=x/(x-1) => y(x-1)=x => xy -y = x => xy-x = y => x(y-1)=y => x= y/(y-1)

So inverse(g(x)) = g(y) :D

Set y = x/(x-1)

y (x-1) = x

yx - y - x = 0

( y - 1)x - y = 0

x = y / (y - 1)

Hence, the inverse is

y= x/( x-1), which is the same as the original.

Let y = x/(x-1)

Switch x and y

x = y/(y-1)

solve for y

(y-1) x = y

xy - x = y

y(x-1) = x

y = x/(x-1)

f^-1(x) = x/(x-1)

Note:

g(x) and its inverse are the same.