1. I don't like your notation, f is defined as a function on real numbers,

so what f(f)(x) means is debatable. I prefer the notation (f o f)(x) or f(f(x))

for composite function.

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

so:

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

let y=x/(x-1), then:

f(f(x))=f(y)=y/(y-1)=[x/(x-1)]/[x/(x-1)-1]=[x/(x-1)]/[(x-(x-1))/(x-1)]=x

RonL