Yes, a function, f, is "injective" if and only if only when . You are correct that, for this example, means that which leads to and then either or . Because of the second possibility, this function is not injective.
As for "surjective", that means that for any y, there exist x such that f(x)= y. To show that this function is NOT surjective, it is enough to demonstrate a number, y such that there is NO x that makes . And one way to do that is to note that is never negative so that is never positive and, finally, that is never larger than 3. A very simple way of showing that this function is not surjective is pick y larger than 3, 4, say, and show that has no solution.
The very simplest way of seeing that this function is neither injective nor surjective is to observe that its graph is a parabola, opening downard, with vertex at (0, 3). Do you see why that immediately show the function is neither injective nor surjective?