DSLR Camera and lens compatibility?

I have a Nikon D5200

I was looking at the Nikon Nikkor IF 24-120mm f/4-22 SWM Standard Zoom Lens
It says it is AF-S on Silent mode, which I know AF-S is necessary for the D5200.

Question is, will this lens be compatible and work well with the D5200?

It is an FX lens, but it says it is compatible with DX as well.
I just want to know I'm not wasting money and dampen the benefit with some non-compatible qualities.

Nikon cameras that can use AF-S lenses, can use either DX or FX lenses, unlike Canon in which the EF-S lenses (DX Nikkor) can't be used on their full frame cameras. The EF-S lenses will damage the mirrors on any of the Canon full frame lenses

It would be compatible. However, you would be paying a price in money and weight putting an FX lens on a DX body. The 18-140mm AF-S VR DX would be a better buy.

It's a Nikon AF-S lens… Why would it NOT work?

As far as the utility of the lens for you… Only you know. I carried the original 24-120 non-Af-S model around on a D100 as my everyday lens, worked great for me. I do tend to work at the longer end though, so not missing a lot at the wide end.

If you have the 18-55 or something even wider like the 10-24, keep that handy for when the 24mm end is just too narrow.