Given the function:
\[f(x) = \sin^{-1}(x - \sqrt{x}).\]
Step 1: Analyze \( g(x) = x - \sqrt{x} \)
The domain for \( x \) is limited to \( [0, 1] \).
Let:
\[g(x) = x - \sqrt{x}.\]
Compute the derivative of \( g(x) \):
\[g'(x) = 1 - \frac{1}{2\sqrt{x}}.\]
Find critical points by setting \( g'(x) = 0 \):
\[1 - \frac{1}{2\sqrt{x}} = 0 \implies \sqrt{x} = \frac{1}{2} \implies x = \frac{1}{4}.\]
Evaluate \( g(x) \) at critical points and endpoints:
\[g(0) = 0, \quad g(1) = 1, \quad g\left(\frac{1}{4}\right) = -\frac{1}{4}.\]
The range of \( g(x) \) is:
\[\left[-\frac{1}{4}, 1\right].\] Step 2: Apply \( \sin^{-1} \)
The function \( \sin^{-1}(g(x)) \) maps the range of \( g(x) \) to:
\[\left[ \sin^{-1}\left(-\frac{1}{4}\right), \sin^{-1}(1) \right].\]
Substitute the values:
\[f(x) \in \left[ -\sin^{-1}\left(\frac{1}{4}\right), \frac{\pi}{2} \right].\]
Thus, the range of \( f(x) \) is:
\[\left[ -\sin^{-1}\left(\frac{1}{4}\right), \frac{\pi}{2} \right].\] Final Answer:
\[\boxed{\left[ -\sin^{-1}\left(\frac{1}{4}\right), \frac{\pi}{2} \right]}\]