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Right, if you load up that 3,000 watt inverter beyond the capacity of the alternator, the battery voltage will lower and so the inverter will be getting power from both the alternator and battery. At some point the battery voltage will get too low and the inverter will stop itself, and the lead-acid battery would incur some wear. Make sure to charge it up afterwards.
One other thing is that alternators often get too hot if they are trying to keep voltage up at the battery as hard as they can ("Full Field") unless you rev up the RPMs so the alternator can cool itself.
Invest in a cigarette lighter volt meter so you can figure it out
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