What is the inverse of the function f(x)=(x-1)/(x-2)

Answer 1

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

What you need to do is solve for x, and then swap x & y:

x-1 = y*(x-2)

x-1 = xy - 2y <<< simplified RHS

x-xy = 1-2y <<< subtract xy from both sides, add 1 both sides

x(1-y) = 1-2y <<< factor LHS

x = (1-2y)/(1-y) <<< divide both sides by 1-y, note: y ≠ 1

Now swap x & y:

y = (1-2x)/(1-x) = (2x-1)/(x-1) ===> f^-1(x) = (2x-1)/(x-1), x ≠ 1

Check:

f^-1(f(x)) should give you x (it does, try it... just plug in "(x-1)/(x-2)" for "x" in f^-1(x) and simplify )