Car Remote control not working from distance

Have you checked the fuse under the bonnet? 15 Amp fuse marked in red on the photo below.

Mitch, No fuse for the headlamps. Controlled by the IPM,(Intelligent integrated Platform Module) fuse box under dash, has a built in smart circuit breaker that trips. It should reset if the key is cycled, but if the short is still there it will open again.
Check the attached links, last link shows IPM.
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check the wires to your amp.may be a broken earth wire.

they might have lost their program. A dealership can reprogram the fobs and it should work. But if they constantly loose program it is the remote module. it is located in the dash under the radio. you'll see an antenna coming off it.

not rechargeable,you must replace the battery

The 01 Optima had the keyless entry buttons right on the key itself. I'm not sure if the LX had the option... it was standard on the SE version. In any case, programming is done at the dealership, and is expensive. They use their special "code reader / writer" that plugs into the OBD II port under the dash. You provide all keys and they are recoded to the module that is located under the stereo. A cheaper option might be an aftermarket alarm installed at a local car audio shop. They are less expensive than you might think, even installed. Good luck.
P.S. the 01 Optima remote key had a problem with very short range. You needed to be within a few feet of the doors for it to work.

if its a factory remote you have to check your deal if the vehicle is doing all those function you may have a bad module on one of those function

all vehicles now need the manufacturers scan tool to program remotes that way if you no longer have remotes or lost them they can remove these to prevent theft but it isnt ther car that essentially loses programming its the remote if you can change batteries soon enough you wont have this concern

you have to get a kia keyless remote a mitsubishi won't work