Faze, as moorche says, means to upset. If people or things are fazed by something, it means that their performance may be affected negatively. You usually hear "faze" in the negative form. "The high wind didn't faze the golfer." I found the following headline: "Brief Federal Shutdown Wouldn't Faze Medicare Or Medicaid" - Neither Medicare or Medicaid would perform badly because of a brief federal shutdown. Do NOT get "faze" mixed up with "phase" (a stage or cycle). They definitely don't mean the same thing. As for being formal, the only place I wouldn't use it would be a scholarly article. For any news item or magazine article, you would see it used freely. The people who read scholarly articles are stuffy anyway.

I would say it is informal. Check yourself: http://www.urbandictionary.com/define.php?term=faze

Yes.............it means.........to upset.