$7 \cdot 119$ If $F\left(\frac{1-x}{1+x}\right)=x$, then which of the following equations is correct?
(A) $F(-2-x)=-2-F(x)$.
(B) $F(-x)=F\left(\frac{1+x}{1-x}\right)$.
(C) $F\left(x^{-1}\right)=F(x)$.
(D) $F(F(x))=-x$.
(China High School Mathematics League, 1984)
[Solution] Let $\frac{1-x}{1+x}=y$, then $x=\frac{1-y}{1+y}$,
$\therefore \quad F\left(\frac{1-x}{1+x}\right)=F(y)=x$.
By the given condition, $F(y)=\frac{1-y}{1+y}$.
$\therefore \quad F(-2-x)=\frac{1-(-2-x)}{1+(-2-x)}=-\frac{x+3}{x+1}$.
And $\quad-2-F(x)=-2-\frac{1-x}{1+x}=-\frac{x+3}{x+1}$.
Therefore, the answer is $(A)$.