I've had my car for about 5 months now. Just the other day it overheated on me & I went to fill up the radiator and read I should check the level on my coolant recovery tank. I downloaded the manual but still can't find it, I don't believe my car has one.. I've attached a picture, hoping someone can spot it. If it's really not there, how bad is this?

 

https://imgur.com/a/U1lOR

 

It certainly looks that way. Something is up. You should have a coolant overflow bottle hanging off the strut tower behind the fuse panel. If you look at the item I marked in red near the radiator cap, that should be where the hose is that goes to the overflow bottle. Is that a hose, if so, where does it go? Also, another item that concerns me, the radiator cap is NOT a factory cap. It's a aftermarket cap with a lever to allow you to release pressure (hence why I think something might be up)....

 

Thank you very much for your speedy reply, I'm new to owning a car and absolutely appreciate it.
From what I can see, I think it might be a hose? It seems to be broken / distorted at the end but it leads no where, just hangs down :/. I've found a coolant recovery tank on the autozone website for $35 so ill be purchasing that piece immediately. Could I have done irreparable damage to my car by driving it like this for so long? (No telling how long it's been like this considering I bought it like this apparently). Should I also replace the weird cap as well?

 

I would replace the Cap, also check your hoses to make sure they are good and not spongy.
I am attaching a couple of photo's for reference on the engine bay.

*Another critical think to check is your Transmission fluid level, and make sure the fluid is RED, if it is brown have it serviced.
When you have a car with disc brakes do not over tighten the wheel lugs or it will warp the rotors and you will have a nasty shimmy in the car when trying to stop. You might want to find a Torque wrench and keep it in the car.
Hope this all helps!

 

If that hose is open ended, then any coolant that would go to the overflow is just exhausting out out of the car. This also means, the car is going to pull in air instead of coolant when it needs more from the overflow.

When you said it overheated, did the car come to a full stop/stall? Did the gauge just go into the red/hot zone? Did an over temp indicator light up on the dash? A piece of advice given to me long ago, if you overheat a car and it still runs afterwards, get the oil changed ASAP. The over temp can negatively affect the properties of the oil, fresh oil is worth while. But no one can predict the long term (if any) effect on what happens.