You are not yet in Complex Numbers. This means that stuff under a square root MUST be Positive or Zero.
So, 1 - x^2 >= 0 or 1 >= x^2 so |x| <= 1
Similarly, 9 - x^2 >= 0 or 9 >= x^2 so |x| <= 3
Clearly, then, the answer key is simply wrong. It should be [-1,1]. Try x = 2 and prove the answer key wrong!