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1.
Face-out is a combination of Facebook and fake-out. It is used to describe someone who likes a person's Facebook status, waits for acknowledgement for liking it or long enough for the person to get a notification for it, and then purposely unlikes it. Usually, public shame and ridicule towards the recipient follows.
Sarah (posts Facebook status): I just left my house and I already miss my lil kitty. I luv you Otis!

John likes her status.

Sarah (commenting on her own status): Hey John! haha isnt he just like the cutest?

John immediately unlikes her status.

John (responding to her comment on status): Face-out! Seriously, nobody cares.

Other common forms: faced-out as in, "you jus got faced-out!"
by The Revolutionist October 03, 2011
 
2.
The action of asking someone out via facebook.
Mike just face-outed Katie, they're going to the movies tomorrow.
by Cesco13 October 25, 2010