Let f(x)=x−2x+2f(x) = \frac{x-2}{x+2}f(x)=x+2x−2. Determine f2024(x)f_{2024}(x)f2024(x) where fnf_nfn is the nnn-th iterate.
x−2x+2\frac{x-2}{x+2}x+2x−2
xxx
x−4x+4\frac{x-4}{x+4}x+4x−4
2x−1x+2\frac{2x-1}{x+2}x+22x−1