Compute the domain of the real-valued ntion f(x)=sqrt(1-sqrt(2-x). Thank you!

Answer 1

since the domain of √u is u>=0, we need

2-x >= 0
That is, x <= 2

Next, we also need 1-√(2-x) >= 0, so
√(2-x) <= 1
2-x <= 1 means x >= 1
-(2-x) <= 1 means x <= 3

So, we have a final domain of 1 <= x <= 2

The graph is at

http://www.wolframalpha.com/input/?i=%E2%88%9A%281-%E2%88%9A%282-x%29%29

You can see that f(x) has a zero imaginary part only in the interval [1,2]. That is the domain of f(x) for real values.