(1) Given $f(x)=\frac{x+1}{x-1}(x \in \mathbf{R}$ and $x \neq \pm 1)$, which of the following equations always holds?.
(A) $f(x) \cdot f(-x)=1$
(B) $f(x)+f(-x)=0$
(C) $f(x) f(-x)=-1$
(D) $f(x)+f(-x)=1$
(1) A