If $f(x)=(x-1)/(x+1)$, then $f(f(x))$ equals:

Substitute $f(x)$ into itself: $f(f(x))=((x-1)/(x+1)-1)/((x-1)/(x+1)+1)$. Multiplying the top and bottom by $(x+1)$ gives $((x-1)-(x+1))/((x-1)+(x+1))=(-2)/(2x)=-1/x.(x$ would mean $f$ is its own inverse, which it is not; $(1-x)/(1+x)$ is just $-f(x)$; $1/x$ drops the minus sign.) Check at $x=3:f(3)=1/2$ and $f(1/2)=(-1/2)/(3/2)=-1/3$.