Understanding the Concept:
The domain of a function is the set of all real values of $x$ for which the function is defined.
In the given function,
\[
f(x)=\sin^{-1}(\sqrt{x-1}),
\]
two separate conditions must be satisfied:
The expression inside the square root must be non-negative.
The input of $\sin^{-1}(x)$ must lie between $-1$ and $1$ inclusive.
Therefore, we solve both conditions carefully.
Step 1: Applying the square root condition.
For the square root to exist,
\[
x-1\ge0
\]
Adding $1$ on both sides:
\[
x\ge1
\]
So the function is defined only when:
\[
x\in[1,\infty)
\]
Step 2: Applying the inverse sine condition.
For $\sin^{-1}(u)$ to exist,
\[
-1\le u\le1
\]
Here,
\[
u=\sqrt{x-1}
\]
Therefore:
\[
-1\le\sqrt{x-1}\le1
\]
Since square roots are always non-negative, the lower inequality is automatically satisfied.
So we only need:
\[
\sqrt{x-1}\le1
\]
Step 3: Solving the inequality.
Squaring both sides:
\[
x-1\le1
\]
Adding $1$:
\[
x\le2
\]
Step 4: Finding the common interval.
From Step 1:
\[
x\ge1
\]
From Step 3:
\[
x\le2
\]
Combining both:
\[
1\le x\le2
\]
Hence, the domain is:
\[
\boxed{[1,2]}
\]