Did the bicycle serve its purpose?
No object serves no purpose, though objects serve a purpose when they are needed. An object that serves no purpose is one that is not needed at the time; however, there would come a time when it did serve a purpose. Synonyms for something that serves no purpose are superfluous, unnecessary, needless, or excrescent.
If you are referring to the phrase "To serve a purpose" then it would mean to give a use for. For example, A pencil holder cup serves a purpose to a pencil... the pencil give the holder purpose and that purpose is to hold the pencil. Basically it means, that something has a use... there is a reason/intended use for this thing.