Question

Let f(x) = x+2/x-1. Find a function g(x) so that f(g(x)) = x.

How do I solve this? I’m stuck on what to do.

Thank you!

Answer 1

g(x) = y

y + 2 / y -1 = x

y + 2 = xy - x

y - xy = -x - 2 ... y (1 - x) = -x - 2 ... y = (x + 2) / (x - 1)

Answer 2

Well, that can only be true if g(x) is the inverse of f(x).

let y = f(x) = (x+2)/(x-1) , I assumed those brackets were needed
then the inverse would be
x = (y+2)/(y-1)
xy - x = y+2
xy - y = x+2
y(x - 1) = x + 2
y = (x + 2)/(x - 1)

<b>g(x) = (x+2)/(x-1)</b>

check:
let x = 7 , or any arbitrary number you want
g(7) = 9/6 = 3/2
f(3/2) = (3/2+2)/(3/2-1)
= (7/2) / (1/2)
= (7/2)(2/1) = 7

It is highly unlikely that I would have obtained that result had my g(x) answer been incorrect.