If air in a turbofan engine bypasses the jet engine after going through the turbofan, why does the jet engine even exist in the system?
The turbofan engine has a giant fan at its inlet, with a much smaller turbojet engine behind it. That turbojet engine is what provides the mechanical work needed to turn the giant fan. About 5 to 10% of the air passing through the fan then passes into and through the turbojet behind it. The turbojet uses this air for its operations: compressing it, burning fuel in it, and then extracting most of that hot air’s energy as rotational work. This rotational work is used to power the giant fan.