whats the job of an overdrive in a lexus LS 400,should l ve it on at all times

If you leave it on and drive in excess of 45 MPH at any time, then you will get better fuel economy.

It is a higher (lower? l forget whether it is the engine or the transmission output that goes first) gear ratio, greater than 1 to 1, which allows ur engine to turn less than 1 time for each time that the drive shaft turns(If it is not on the other side of 1 to 1 from the other gears, it is not overdrive.) Basically ur engine does not have to run as fast for u to go at higher speeds, so u use less fuel.

Leave it on unless you're driving up hills, or in some situation where the transmission can not decide what gear to go into & shifts a lot or shudders.

overdrive in any vehicle is like having that extra top gear. you are cruising along on the freeway and instead of the engine being at say 3500 RPM being in top gear, overdrive gear allows it to kick down to say 1500 RPM yet maintain the speed. Less wear on the engine, less gas used, less emissions.

Think of it like normal cars are all 10 speed bikes...but when you add overdrive, its now a 12 speed bike.

Overdrive reduces engine RPMs at higher speeds and improves mileage. You can leave it ON all the time unless you want to have engine braking, as when going down a hill.