As for this, there is a wrong solution ahead, don't be misled.
You can find the first few fn(x) to find the pattern.
f 1(x)=( 1+x)/( 1-x),
F2(x)= f[f 1(x)]=( 1+f 1(x))/( 1-f 1(x))=- 1/x,
f3(x)=( 1-x)/( 1+x),
F4(x)= x;
……
It can be deduced that f(4k+ 1)(x)=f 1(x),
f(4k+2)(x)=f2(x),
f(4k+3)(x)=f3(x),
f(4k+4)(x)= F4(x);
therefore
f 2006(x)= f(4 * 50 1+2)(x)= F2(x),
That is, f2006 (x) =-1/x.