Problem 166

Since $f(x)=x^{2}-1$,

We have $f(±1)=0$ and $f(± \sqrt {2})=1$.

Therefore, the possible domains $D$ are: $\{1, \sqrt {2}\}$, $\{-1,- \sqrt {2}\}$, $\{-1, \sqrt {2}\}$, $\{1,- \sqrt {2}\}$, $\{-1,1, \sqrt {2}\}$, $\{-1,1,- \sqrt {2}\}$, $\{1, \sqrt {2},- \sqrt {2}\}$, $\{-1, \sqrt {2},- \sqrt {2}\}$, and $\{-1,1, \sqrt {2},- \sqrt {2}\}$, making a total of $9$ cases.

Hence, the answer is $\boxed{9}$.

To solve this problem, we inferred potential $x$-values in the domain based on the given function expression and the values in the range. Then, we combined these $x$-values to find the various possibilities for domain $D$. This question primarily assesses your understanding of functions and the ability to make simple deductions, making it a foundational problem.