Troubleshooting a possibly defunct low gas / low fuel indicator light

Just as the title suggest, I'm in the process of troubleshooting an issue with my 97 Pathfinder SE. A while ago, I had an issue with my spark plugs. This was maybe 3 or 4 months ago - one of the spark plugs was misfiring, so I went ahead and changed all the plugs, the wires, the rotor, ... in a nutshell, I did my own tune up. However being new to D-I-Y auto repair, I wasn't certain what the issue was until I had some friends point me the right way. One thing I did notice while this was going on was that a lot, if not all, of the indicators lights were acting bizarre - it reminded me of all those movies you'd watch where a plane cockpit dashboard lights would go crazy as the plane was crashing. All is fine now with the exception of the low gas / low fuel indicator light.

Since then, I've only seen it illuminate once. Usually it would brighten up when I was near empty, but well before I would run completely dry and could probably squeeze out about another gallon of gas, give or take a quarter-gallon before being totally stranded.

As I've been driving more recently to test and troubleshoot the indicator light, I have to believe that it's not working, because I've come close a few times since then to being bone-dry. The symptoms were pretty much what you could guess - the car driving in a way where you could tell it was going to shut off for lack of gas.

So, my question for anyone who can help me is what route should I take in trying to deduce what the issue is causing this problem? I'm guessing I should check fuses first, then perhaps check the back panel of my instrument display? Yes? No?

While it's not a critical thing since the fuel gauge works just fine and I use the trip odometer to help me figure out when I should refill on gas, I'd still like to know that the light is working.

Any help is appreciated.