6. 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. $F(F(x))=-x$
Let $t=\frac{1-x}{1+x} \Rightarrow x=\frac{1-t}{1+t}=-1+\frac{2}{1+t} \Rightarrow F(x)=\frac{1-x}{1+x}=-1+\frac{2}{1+x}$ $\Rightarrow F(-2-x)=-1+\frac{2}{-1-x}=-2+1-\frac{2}{1+x}=-2-F(x)$, option $A$ is correct; $F\left(\frac{1+x}{1-x}\right)=-x \neq F(-x)$, option $B$ is incorrect; $F\left(x^{-1}\right)=-1+\frac{2}{1+x^{-1}}=-1+\frac{2 x}{1+x} \neq F(x)$, option $C$ is incorrect; $F(F(x))=F\left(\frac{1-x}{1+x}\right)=x \neq-x$, option $D$ is incorrect. Therefore, the correct choice is $A$.