Battery backup (UPS) for computer while on house batteries?

So would the inverter be connected to power the 110 outlets all the time, regardless of what the power source is? Such as shore power or generator only powers batteries and inverter always pulls from batteries to supply 110 to RV? In the case, switching would be a non-issue, obviously. If that's the case, would that shorten the life of the batteries themselves?

Thanks to everyone for your replies thus far!!