Are you defining sqrt(x²) to be +x and -x? If you are, then sqrt(x) is indeed not a function.

You could make it a function by defining sqrt : N -> Z×Z, so sqrt(4) = (-2,2), and we're back to the property that any value in the domain (the natural numbers) is mapped to a single value in the co-domain (a unique pair of integers). Of course, this is not the traditional sqrt function that people normally use, since it doesn't obey the property (sqrt(x))² = x (since obviously taking the square of a pair of integers will not return an integer, for usual definitions of squaring).

Note that a bijective function has both this property and the opposite, that for any value in the co-domain there is also a unique value in the domain that maps to it. Sqrt, both in the traditional definition where it only returns positive roots, and in this definition where it returns a pair of roots, is actually also bijective. Squaring is a non-bijective function, though.