Since I do not have a phone line anymore I use faxzero when I need to send a fax. Although there are a few draw backs
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Most faxes from the past decade are CCITT Group 3 or 4, containing image/g3fax or image/g4fax data which is encapsulated, or rather modulated, in one of V.27, V.29, V.17, V.34, or V.34bis.
Note that the fax transmission is not so-much a transmission, as a 2-way communication. Thus, you cannot simply play this audio back into a fax machine and expect it to work. However, luckily, the fax image data transmission itself is uninterrupted, so it is possible to filter out the unnecessary data.
What you will need to do is demodulate the content to retrieve the binary fax session and to filter out and extract the image/g3fax or image/g4fax data.
The late Tony Fischer wrote a software fax demodulator for V.34. It only works on Irix, so you'll need to port it. There was a Linux port at some time, but I couldn't find any valid links. http://www-users.cs.york.ac.uk/~fisher/software/modem/current/
There is also the Linmodem project.
Between these resources and some C programming, you should be able to extract the faxes... this isn't for the faint of heart, but possible, if the data is that important to you.
Best Answer
If you want to send/receive faxes, then just connect the machine to the phone line. The phone company does not sell/lease "fax" lines or numbers.
WiFi is for networking devices, not phone equipment. If you connect this HP by Wifi, then you could use the printer/scanner as a wireless network printer. Sending/receiving faxes from/to the PC using the HP over the phone line would depend on software installed on the PC and how the HP is configured (e.g. would received faxes be printed (on paper) immediately or sent (softcopy) to the PC?)
Addendum:
The fax is a phone device/service. If you choose to use the "online service", then the fax capability of the HP is totally irrelevant. The Internet and your PC with the "online service" replace the fax machine at your location.
BTW most businesses have been able to accommodate me when I ask them to exchange e-mail attachments instead of faxes. Scanning the document(s) and then composing the e-mail is more work then sending a fax, but it is doable when your faxing requirements are so infrequent.