Originally Posted by Timothy Miller
The FN + F-keys are hardware keys. It tells the BIOS/EUFI to switch from display 1 to display 2, all the OS is needs to do is be able to detect it. However, not all models do they work with all outputs. I'm guessing there's probably something in the xorg.conf that's preventing it from displaying, but I'm not sure what could cause it.
lspci was just so I could see the full hardware listing of what your laptop had, it will not tell you which is being used, since they're both seen by your OS at all times.
timothy brings up a good point. this is os agnostic hardware related. so from a cold boot do you you see the manufacturers splash screen or even the grub menu on the tv (before the os is launched) ? if not there mite be something in the bios that has to be toggled.
other than that i would try a different tv, cable and laptop to isolate which is faulty.