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

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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...

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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].

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