Question

Find inverse function of f(x) = (x-2) + 1/(x-2)

Answer 1

first, put on common denominator: f(x)=(x-2)^2+1 / (x-2).
(x-2)f(x)=(x-2)^2+1
xf(x)-2f(x)=x^2-4x+5
0=x^2+(-4-f(x)x+(5+2f(x))
Now, you got a quadratic with a=1,b=-4-f(x) , c=5+2f(x)
Solve it. Now, you will get some roots. But you know that by the fundamental thm of algebra, any complex polynomial of degree n can be written in the form (x-root1)...(x-rootn). You will thus get somehting like x=4+f(x)+-sqrt(f(x)^2-4) /2, which is what wolframalpha gives https://www.wolframalpha.com/input/?i=inverse+f(x)+%3D+(x-2)+%2B+1%2F(x-2)