The full form of fax is “telefax” or “telefacsimile.”
From “facsimile” to “telefacsimile”
“facsimile” has traditionally meant “an exact copy or replica” – the term has been used in the english language since the 16th century.
The prefix “tele” translates to “over a distance.”
Inventions from the late 1800s and early 1900s frequently used the prefix “tele” to denote their operation.
Examples still around today are “television” (vision over a distance) and “telephone” (sound over a distance).
There were also inventions similar to fax called “telautographs” (signatures) and “telephotographs” (photographs) that sent messages over telegram wires.
The “telefacsimile” was a logical word for the function that a fax machine does.
From “telefacsimile” to “telefax”
The word “telefax” appears to have entered language very soon after “telefacsimile.”
It is likely a shortened version of “facsimile” as the “c” in “facsimile” is pronounced like a “k.”
From “telefax” to “fax”
The first use of the term “fax” is difficult to find as so many publishers have listed their fax number in every book they publish. Even for older books, many have been republished, making searching for the term “fax” difficult.
“Fax” was likely a logical next step from “telefax” the same what that the word “phone” evolved from “telephone.”
Facsimile evolves as a word
Since the introduction of fax machines, the definition of “facsimile” has evolved to include both its traditional definition (of an exact copy of a document) to also include a secondary meaning that is exactly the same as fax.
Many acronyms have been mentioned for the meaning of “fax.” Most of them involve Xerox, however most Xerox faxing equipment wasn’t in use until much later. Acronyms such as “far away Xerox” or “facimile automated Xerox” are likely Backronyms, or acronyms applied to an existing word after it had already entered the language.