offensive : causing someone to feel resentful, upset, or annoyed.
(Or....insulting, rude, derogatory, disrespectful, hurtful, wounding, abusive, objectionable, displeasing, annoying,provoking, humiliating, discourteous, uncivil, impolite, unmannerly, unacceptable, shocking, scandalous, outrageous)
From Oxford Dictionary.
So, yes I do feel upset and annoyed when someone uses a term which is racist, sexist, disabilist or homophobic, because it is hurtful and abusive - surely that is exactly what offensive and being offended means.