The domain of the function $f(x) = \sin^{-1}(\sqrt{x - 1})$

For the function $f(x) = \sin^{-1}(\sqrt{x - 1})$ to be defined, the argument of the inverse sine function must lie in the interval $[-1, 1]$.

$-1 \le \sqrt{x - 1} \le 1$

Since the principal square root is always non-negative, this inequality reduces to:

$0 \le \sqrt{x - 1} \le 1$

Squaring all parts of the inequality, we get:

$0 \le x - 1 \le 1$

Adding $1$ to all parts, we obtain:

$1 \le x \le 2$

Thus, the domain of the function is $[1, 2]$.

Answer: $[1, 2]$