I know how to invert simple functions and how to solve simple radical equations, but this problem stumps me:

find the inverse of y = f(x) = sqrt(x) / (x-1)

Thank you.

Try this: square your original equation to get:

y^2 = x/(x^2-2x+1)

Multiply through:
y^2x^2 - (2y^2+1)x +y^2 = 0

Apply the quadratic formula to solve for x, and you get:
x = (2y^2 +1 +/- sqrt(4y^2+1))/(2y^2)

Now check: in the original equation if x = 4, then y = sqrt(4)/(4-1) = 2/3. Plug 2/3 into the above for y and you get x = 4 or 1/4, so it works.