hello, trying to revive a neglected tacoma

I just changed plugs and coils, I cleaned ("rebuilt") fuel injectors and in this process I noticed a cracked hose. Im having trouble identifying said hose and was hoping someone could help. So the large hose that runs out of the driver side of the radiator hooks to ?? not sure what the part is called .. its located under the power steering reservoir ??? the hose that comes out of this unnamed part is broken at the device.... what is this thing called and where does that hose go/what is it for??? Any advice will be greatly appreciated. Im a motorcycle guy and while mechanically inclined in that world im not quite there, ha, in truckland just yet. anyway.. thanks in advance

 

Pictures, and vehicle year and engine.

 

Its a 2.7 '09. Pics to follow

 

so, big guy in middle of pic comes from drivers side of radiator. i put metal tape around broken hose in question to show what im talking about. this is directly under power steering

 

Well since there is only an upper and lower radiator hose by both and your good to go.

 

the hose coming out of the part under the fuel pump is broken

 

Now I see. Thought your were talking about the rad hose. My bad.