What is a functional group?
Your books statement that "carboxylic acid is a functional group" doesn't seem quite right to me. I think it's better to say that a carboxylic acid is an organic molecule containing a carboxyl functional group.  

Similarly, I wouldn't say that an alcohol is a functional group, but that an organic molecule containing a hydroxyl group is an alcohol.  

Of course you can have single large molecules with multiple functional groups that could be called, at once, for example, an alcohol, a ketone and a carboxylic acid, because it contains a hydroxyl functionality, a carbonyl functionality and a carboxyl functionality.  

I hope this addresses what you were asking, don't hesitate to ask if I missed something.