Am reading at wiki, which states 4 examples to state the 3 properties of a function or their composition(s).
I am confused over the 4th function $f_4$ which is stated to be surjective.
As per my knowledge, a surjective function is one that maps all the co-domain elements. So, a map over the positive reals given by $x \to x^2$ would be not covering all the elements of the co-domain. Or may be I am wrong, and the given function is not a surjective function over integers (positive or otherwise), and is surjective over the positive reals as all values can be taken in the continuum.
If my last statement is correct, then why $f_3$ is not surjective. Please give some example, if feel is needed.
$f_4: \mathbb{R}_{\ge0} \to \mathbb{R}_{\ge0}$ with $f_4(x) = x^2$ is surjective because unless $x$ is negative, $\sqrt{x}$ is a positive real number so it covers all the positive real numbers (when we say $f_4(\sqrt{x}) = x$ with the same domain and codomain, it might be easier to see why it is surjective).
$f_3$ is not surjective because notice that codomain is $\mathbb{R}$ and there is no real solution of the equation $f_3(x) = -1$ for example (it has complex solution $i$).