Question:
If $f(a) = \frac{1}{1-a}$, find the product $f^{-1}(a) \times a \times f(a)$.  (Assume $a \neq 0$ and $a \neq 1$.)

Answer:
Substituting $f^{-1}(a)$ into the expression for $f$, we get \[f(f^{-1}(a))= \frac{1}{1-f^{-1}(a)}.\]Since $f(f^{-1}(x))=x$ for all $x$ in the domain of $f^{-1}$, we have \[a= \frac{1}{1-f^{-1}(a)},\]Solving for $f^{-1}(a)$, we find $$1 - f^{-1}(a) = \frac{1}{a} \quad \Rightarrow \quad f^{-1}(a) = 1-\frac{1}{a} = \frac{a-1}{a}.$$So $f^{-1}(a) \times a \times f(a)$ is  $$\frac{a-1}{a} \times a \times \frac{1}{1-a} = \boxed{-1}.$$