how can you tell if the air conditioning compressor is working

You will need a good set of HVAC gauges and the skill to read them. I do not have either. Let the professionals diagnose and you do the repairs if needed.

locate compressor,start vehicle,turn a/c on high,watch a/c compressor,where the belt goes around it,IF THE BELT IS STILL ON THERE,MAYBE IT BROKE AND YOU HAVE NO BELT,IT WENT BYE-BYE.watch the clutch,when the clutch spins,the compressor is running.if the belt is missing,disregard watching compressor,cause it ain't coming on.replace belt.