Just a piece of advice from my experience, you don't want to setup a trust between domains that will not constantly be able to talk to each other. There's a good chance that you'll spend more time breaking and re-establishing the trust then you will doing anything else.
If you are dead-set on doing this and you aren't using WINS at your home, you'll need to create an LMHOST file for each machine that will be contacting your domain at work. In it you will need to make entries for each machine on the distant end that you'll need to talk with.
Are you ever on ICQ, Yahoo! Messenger, AIM, or MSN Messenger? There's a lot to this whole subject and real-time chat would make this a lot easier to talk about.
Email me at majestrate@hotmail.com with the answer/username to my question and a time that you're on. I won't be signing in to any of those until about 1730 PST or 2030 EST.