Good Evening, Is there a way.

Good Evening,

Is there a way you can tell if the water pump or a head gasket is gone?
I just too my car to a repair shop the mechanic stated that it was a water pump, when I picked the car up to take it home 400 dollars later it was doing the same thing. So today they called and stated its a head gasket. So my question to you is, if they would have done further investigation could they could have known if was a head gasket before I replaced the pump?