My gsx 600 f ticks over at around 3000 rpm once warm.

Yea, that is way too high. Thankfully your bike is nice and old so idle adjustment is simple. Look for a knob in between the 2nd and 3rd cylinders carburetors. This should be accessible without removing any bodywork, but I'm not certain with your bike. You should set your idle to something between 1~2k rpm when the bike is fully warmed up. If your carb hasn't been serviced in awhile having it idle at 2k rpm will be more stable.

Hi, Timothy_towd "ABSOLUTELY" you need to go through all 4 carbs set the proper float height and then have them synced for this scenario you will need your service/owners manual if you can't find the first and best tool you ever bought for your Yamaha, despair not, for a mere zero $0 you can download another one. For more information about your issue and valuable "FREE" downloads that you will need please click on the blue links below. Good luck and have a wonderful day.
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turn the cable adjuster back to its original position ie between 600-700 rpm look to the left hand side of the carbs you will see a large adjusting screw on a metal bracket whilst engine is running turn inwards towards rear wheel until between 1000 - 1500 rpm remember to warm engine first as rpm will rise if idle is set with a cold engine as the engine warms up when riding

Must be throttle sticking .
Check , Cable at carb end is in holder ,
Cable is not kinked,
Cable at handlebar end is properly in slide,
Pistons in carbs are free to slide
There are no obstructions in carbs.
Try, Disconnecting cables from carbs ( if your bike idles , its the cables)
Removing air intakes and looking down the carbs with a mirror and torch ( slides should be seating almost closed)

If your bike has ever sat for more than a few months at any time, the carburetors could be plugged with gasoline residue. If so, they will need to be disassembled and cleaned.
Otherwise, make sure all of the normal tune up items have been performed; valves adjusted, air filter, spark plugs, etc. And make sure there are no air leaks between the carburetors and the engine.
Finally, if the above things are all good, then I would perform a compression check. With the throttle held wide open, you should see at least 140psi of compression, on every cylinder. Anything lower than that will cause an erratic idle quality and overall poor performance.

Carbs need to be removed, dismantled and cleaned. Soak in solvent or equivelant for a couple hours and use air nozzle to blow dry. Make sure to blow out jets.

I had a similar problem with my Vulcan years ago, it turned out to be the Ignition Wires. Check to see if you get spark while the bike is hot.

You have carburetor issues my friend, take those things apart and clean them. Probably time to rebuild them if they have never been apart yet.

There should be an idle adjustment located behind one of the side fairings, its usually a black dial mounted to the case somewhere low.

are the carbs balanced correctly and are you sure that all of the jets are clean. if not try it, it might work. sluggish throttle response is normally to do with the carbs.