I'd hate to ask questions without searching but I don't even know the name of this, or what it does :P.

But the small hose, right before the throttle body on the intake, that goes into the engine (near the rear spark plugs), what is that? And if you feel like telling me, what does it do?

I've been experiencing bad economy, and I know vac hoses account for some of that, so I set out on checking them, and this one was cracked (engine side).

Thanks.

-edit-
On another note, would someone mind showing me to the 01 Mitsubishi Diamante (Magna TJ if I recall correctly.) service manual?
I've clicked links but they're gone. OR what you guys use to learn about this car. I'd love to learn as much as I can.

Its a crankcase pressure hose, it vents crankcase pressure back into the intake.

If this was split you may have been getting unmetered air into the intake and this could lead to your poor fuel economy. I had a leak with my EGR pipe once and it caused poor fuel economy and a loss of power as well as setting mt TCL OFF light to come on after the revs hit over about 3000rpm and it would stay on until I turned the car off.

Oh thanks for sure. I did replace it with a smaller hose will that cause any adverse effect? In theory I think it wouldn't.