I'm afraid you've got two different different problems - first, you are having trouble recognizing functional groups, and second, you are having trouble with the structure behind the IUPAC system. Are you just beginning your organic course? I had assumed from the title of your paper (mid-term assessment) that you had already seen most of this, but maybe I got the wrong impression?
Anyway, the suffix "-one" that you are using is the suffix for a ketone, and neither of these two compounds contains a ketone. If you look at the table of functional groups on that organic nomenclature page, you will note that a functional group has the structure
That is, a carbonyl (carbon atom with a double bond to oxygen), attached to carbon atoms on both sides (R groups by definition are attached through a carbon atom). Your problem 3a does not have a carbonyl group at all, your problem 3b has a carbonyl attached to a hydroxy group on one side, which make up a different functional group (which you should be able to recognize - or find on the table).
Second, the "-one" is a suffix - it comes at the end of the word. The major functional group in your compound, in the IUPAC nomenclature, is identified by the suffix. So even if your 3a was actually a ketone (which it is not) that had a 6 carbon chain with the double-bond to oxygen on the third carbon, it would have been 3-hexanone, not 3-onehexane.
Incidentally, I only recommended the wiki page as a way to jog your memory on IUPAC nomenclature. It has been abbreviated to the point where it is useful as a reminder, but would be very difficult to use as a primer on nomenclature. If you have never seen this nomenclature before, it would be much more useful to find a college textbook on organic chemistry and get a more detailed explanation with more illuminating examples to work your way through.