DX lenses are thus marked. Anything else would be FX.
The Nikon F-mount was introduced in 1959, long before digital and DX sensors. They were not marked as full-frame or FX because everything was full-frame back then.... it was all simply 35mm film.
When digital came along and smaller-than-24x36-mm sensors hit the market, Nikon designated them as DX sensors and marketed DX lenses to go along with them, using the DX monikor to designate the lens as having a smaller projected image circle. The older full-frame lenses could be used too, but they weren't marked as such. The tradition continues today.
If it doesn't say "DX" on it, it's full frame. Of course, that's just for Nikon. All bets are off for 3rd-party glass.
So the lens would be usable on both DX and FX bodies.
Whether you would benefit from jumping aboard the FX boat depends on what you shoot, how you shoot, and what your needs are.
FX is not a goal, it's a tool. Buy the tool if you need it, but don't buy a bazooka to hunt squirrels.