Why is not [-2,0] also in the range of sqrt(4 - x^2) along with [0,2]?

Hello!
When we define the square root function, the function which we start with is a square function (y=x^2).  We want a function which, given an y, would return an x such that x^2=y.  In other words, we want the inverse function for the square function.
But the problem is that the square function gives the same result(s) for x and -x: x^2 = (-x)^2. Therefore actually for any positive y there are two roots: positive and negative. Usually we want that any function be one-valued (it is more convenient).
For this reason it was agreed between mathematicians that the symbol sqrt(y) will denote the positive (non-negative) root, at least for real numbers. And in our example, sqrt(4-x^2) is also non-negative for all x's for which it is defined (-2lt=xlt=2). This is the cause why [-2,0) isn't in the range of sqrt(4-x^2).
Note that the domain of this function does contain [-2,0) along with [0,2].