The first fax machine was patented in 1843, several years after the telegraph. Mary Bellis, Alexander Bain and the First Fax (thoughtco.com). Ms. Bellis wrote:
Faxing is by definition a method of encoding data, transmitting it over a telephone line or radio broadcast, and receiving a hard copy of the text, line drawings, or photographs at a remote location.
The technology for fax machines was invented a long time [ago]. However, fax machines did not become popular with consumers until the 1980s.
“At the beginning of the 20th century, the AT&T Corporation advanced fax technology by sending photos via wire transmission. Soon after, The Radio Corporation of America (RCA) successfully transmitted a wireless fax across the Atlantic Ocean. It wasn’t until decades later though that the first modern-day, commercialized fax machine was introduced by the Xerox Corporation. Over the past 75 years, fax transmission time has reduced from an average of six minutes to an average of one minute. The classic ‘fax machine’ of the 1980s (which cost $20,000 in 1982!) has also become obsolete, and faxes are now routinely sent from computers, smartphones or tablets. ” When Was the Fax Machine Invented – eFax
 Bellis, Mary. “History of the Fax Machine.” ThoughtCo, Aug. 28, 2020, thoughtco.com/history-of-the-fax-machine-1991379.