Is a modem required in order to use a fax machine and send/receive faxes?

My desktop dimension 2350 computer of 8 years died (it had the phone jack modem hookup). I had a fax/scanner/copier hooked up to it that did well, until I got rid of the cable company phone service, and got that magic jack phone service.

Aside from that I have been computer shopping (desktop) off and on for the last several months, and have looked away from computers without modems, because I want my ability to fax back. I know that some have options to get a modem, but not all of them do.

I also don't want to have to buy a separate modem unless I absolutely have to, and I don't want to get one from the Cable company. I would like to have my own, preferably on the computer, if that is what I need to do in order to fax.

I have heard there is a wireless modem? What would I need on a computer (like slots, ports, cards, etc.) or what equipment would I need extra? Is this the new way of being able to fax?

I would appreciate any valid input. Thanks!