Question

domain of arcsin(x-2sqrt(x))?

Answer 1

The domain of the inverse sine function is \([-1,1]\). In other words, given \(\arcsin f(x)\), the domain is
\[-1\le f(x)\le 1\]
where \(f(x)=x-2\sqrt x\).

Answer 2

How would you start after that? I suppose we separate \[-1\le x -2\sqrt{x}\] and \[x - 2\sqrt{x}\le1\]
What then? Do I raise everything to the second power?

Answer 3

If you consider a substitution \(t=\sqrt x=x^{1/2}\), you can more easily see that this is a quadratic inequality:
\[x-2\sqrt x-1\le0~~\iff~~(x^{1/2})^2-2x^{1/2}-1\le0~~\iff~~t^2-2t-1\le0\]
You'll solve in terms of \(t\), then back-substitute to find the solution in terms of \(x\).