well, no you don't simply "plug in the values of the endpoints" unless your function is non-decreasing/non-increasing on the interval. as it so happens, 4 - x^{2} is decreasing on [1,2], so in this instance you can do that. f(1) = 3, and f(2) = 0, so the image f([1,2]) = [0,3] (we have to reverse the order of the endpoints because f is decreasing).
on the other hand, we couldn't do this for [-2,2], since f(-2) = f(2) = 0, but clearly the image of this interval contains more than the point {0}.
i'm not surprised that the book is wrong, since its definition of f(S) is likewise incorrect, it should read:
$\displaystyle f(S) = \{f(s) | s\in S = [1,2]\}$, although perhaps this is a bit nit-picky on my part.