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I posted a photo to show a part that a car I just purchased is missing. What does it do, and will its absence damage the engine?

 
Rep Power: 280

That is the stock (oem) intake filter box. The box serves as a housing for the air filter, which filters dirt and debris from entering the intake manifold. I have seen some stock intake systems have the filter closer to the engine too, and some systems have a big box just for air passage and as a catch can I suppose.

Without this part you are subject to dirt,bugs,sticks,leaves,gravel, etc... getting into your intake piping, then in to the intake manifold, then deeper into the abyss.

wow.

 
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I don't think it's really necessary. To put any aftermarket intake on you have to remove it anyway.
The guy before you probably had an aftermarket intake on and then for whatever reason removed it before selling the car to you

 
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the filter for the intake is actually on top of the throttle body it is probably just louder with that part missing

 
Rep Power: 388

I'm not too famil with your engine, but the stock intakes have a resonator and a filter. Both are boxes. Removing the resonator will make your intake sound a little louder- nothing crazy tho. Removing your intake filter is just dumb because you can get crap in your engine.

 
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Yea, the filterbox is def' on top of the TB w/ filter installed. I guess as long as there's no harm no foul with this other box missing, I won't worry about it too much for now. Though I'm sure I will get a little more heat soak....

 
Rep Power: 245

Triz is correct. That is a resonator. It's an empty box to muffle the sound. There's no filter in it. Your car will just run louder without it.

 
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yeah the filter's in the other box... back in high school I took that piece off (resonator) for months because I liked the sound when I opened the throttle up... your car will be just fine