Did they even have dishwashers then?
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Hey now…You were the guy ahead of me at the counter as I recall…
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Some friends of my parents asked if we were going to get one of those new-fangled automatic dishwasher things. My dad said that we had two already. He said they even responded to voice commands. And then to show their friends how well the dishwashers worked, he looked at my brother and I, and said “Go wash the dishes.”
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Likely from the Dish washer. An image would help.
If from an appliance needs a water trap, IE High-loop.
From a private message from the OP, this is the hose in question. I am more bothered by the duct tape on the tailpiece than the rubber hose. Doesn’t appear to have an AAV either.
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I am too, it should have been FLEX TAPE!
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like stated there are bigger issues with that trap assembly to worry about…
They should have used black electrical (vinyl) tape instead of duct tape for a better look. 
The rubber hose in question for the dishwasher drain will most likely outlast the hose that came with the unit.
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Observation: Poorly assembled kitchen sink drain…
Tail piece not aligned with the trap adaptor, trap.
Trap adaptor not properly secured to the trap union. Coupling installation defects.
Poor trap weir orientation.
Suspect dishwasher tubing dispersal arrangement.
DWV riser pipe appears open and not capped.
Poor overall support.
Plumbing Traps & Trap Defects
Source leaks, water damage, mold, sewer gas smells, septic tank odors
It appears to be a part of the dishwasher drain assembly coming from the air gap using exactly what you described. I would have no problem with it and the rubber hose will likely last a lot longer than the cheap flexible pipe that comes with the dishwasher 
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Brian, doesn’t a high-loop act as an airgap to prevent back siphonage?
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It does prevent back-flow siphon for sure.
The OP’s photo shows a pipe assembly consistent with an air gap device.
Here you can buy it complete with the black hose for a few bucks.
This is a correct statement. The anti-siphon device is very typical in jurisdictions that follow the UPC.
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Plumb is important in plumbing. The non flex drain lines need to be in line, plumb, almost level. Starting with the sink tail piece. The gaskets will work with a bit of askew, but past that limit and they will leak in addition to not being tightened up enough. The addition of teflon tape is of marginal advantage.
There might be a dip in that after the air gap flex line. It will hold water and smelly gunk. It might back up the drain water through the top. Best practice is gravity drain with sufficient slope.
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