$K(x) = x^2.$ The domain and range of this function comprise of non-negative real numbers. If it were real numbers instead of "non-negative" real numbers, then it seems easy to prove it by counterexample. The range includes negative integers but their square root can't be represented as real number. It is complex number and so, the function is not onto function.
But in case of "non-negative" real numbers, it is easy to see that the range will include only positive real numbers (and 0). And intuitively it seems obvious that square root of positive real numbers is another positive real number. So, the function should be onto. However, how do I formally show this statement? I am assuming the intuition I presented in the earlier paragraph doesn't qualify as formal proof.