Step 1: Understanding the Domain of Inverse Sine Function

The domain of the inverse sine function, $\sin^{-1}(u)$, is $-1 \le u \le 1$. This means that for the function to be defined, the argument of the inverse sine must lie within this interval.

Step 2: Applying the Domain Condition to the Given Function

For the given function $f(x) = \sin^{-1}(-x^2)$, we must have $-1 \le -x^2 \le 1$.

Step 3: Simplifying the Inequality

We have two inequalities to consider:

1. $-x^2 \le 1$
2. $-1 \le -x^2$

Step 4: Analyzing the First Inequality

The inequality $-x^2 \le 1$ can be rewritten as $x^2 \ge -1$. Since $x^2$ is always non-negative for any real number $x$, this inequality is always true for all real numbers $x$.

Step 5: Analyzing the Second Inequality

The inequality $-1 \le -x^2$ can be rewritten as $x^2 \le 1$. Taking the square root of both sides, we get $|x| \le 1$, which means $-1 \le x \le 1$.

Step 6: Combining the Results

Since the first inequality is always true and the second inequality gives us $-1 \le x \le 1$, the domain of the function is the intersection of these two conditions, which is $-1 \le x \le 1$.