What are three purposes for satire and three methods used by satirists to achieve their purposes?
I will do the best I can to answer your question, but it sounds like the kind of thing that comes out of a specific textbook (which, of course, I don't have access too : ) I say that because it sounds like the kind of question where three "purposes of satire" were given (probably in bold text) with explanations underneath.
The problem is that I can only think of one purpose for satire, but lots of ways it is handled. The purpose of satire, as far as I know it, is to take the shortcomings of a person, group, or society and make a comment on them using various (often comedic) methods. The idea of the satirist is that these shortcomings need to be addressed and changed and by poking fun at them he/she hopes to change the audiences mind about them.
First, types of satire:
Horatian: This is the kind of satire you see in shows like "The Simpsons" which poke fun at people (and society's) foibles. It means to do so light-heartedly and without really offending too many people. It is common in today's culture.
Juvenalian: This is more of satire "with barbs." It is not meant to be playful or funny but rather to satirize something by showing more directly its underbelly. "Animal Farm" is a good example of this...darkly satirizing society.
The satyrist has many tools at his/her disposal. Sarcasm plays a great part in this type of writing, but exaggeration, juxtaposing the issue with another like it, and "double entandre" are all viable weapons for the writer to make use of.
The main point of satire, though, is to make an "end run" around the audience and get them to accept a message, through sarcasm, parody, or whatever, that they otherwise wouldn't probably consider or accept.