To help clarify things: a generator makes
120V AC power to run your electrical outlets, television, microwave, coffee maker, CPAP machine, etc. An inverter changes
12V DC into 120V AC power to run those same appliances. It gets the 12V power from your battery, so it runs down the battery. The more AC power you use, the faster it will run down the battery. That's where the generator comes in. It will recharge the battery, so it can power the lights, thermostat, and the inverter.
As other posters have said, an inverter under load will run down a single battery pretty quickly. That's why serious boondockers will add a second (or more) battery(ies) to extend the time between required recharges with the generator.
Of course, if you won't be using any 120V appliances while off the grid, your single battery should last a day or two. But if you're not using 120V appliances, you wouldn't need the inverter. Hope that clarifies things.