"Cut off" has 6 definitions - 5 are formal, and 1 is considered informal.
Depending on the definition, some are inseparable and some separable. I also explain subtle differences to other similar words.
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Definitions
1. To remove a part of something, using a sharp tool such as a knife.
["cut off"] + [something]
["cut"] + [something] + ["off"]
"His finger was cut off in an accident and he was fortunate enough to get it reattached."
"I'm going to need some help cutting the branches off."
"Cut" vs "Cut off"
"Cut" is to break the surface or to divide something into smaller parts.
"Cut off" is to remove something completely.
"He cut his finger."
This means he cut the skin on his finger so he needs a band aid.
"He cut off his finger."
This means he completely severed his finger so he's missing one finger now.
2. To stop providing or supplying something.
["cut off"] + [something]
["cut"] + [something] + ["off"]
"Cut off the enemy's supplies and their survival is compromised."
"If you're unable to pay your bills on time, they won't cut your electricity off if you contact them and work out an arrangement."
3. To cause a person or place to become separate, or cause someone to be or feel alone.
["cut off"] + [somebody/something]
["cut"] + [somebody/something] + ["off"]
"He became a recluse and cut himself off from society."
"Some villages were cut off because of the flood."
4. When a phone connection/line is disconnected.
Using the active or passive voice can imply if the disconnection was accidental or intentional.
Passive voice - accidental disconnection
This scenario is beyond the control of either party.
[somebody/something] + ["cut off"]
"My phone is low on battery, the line might cut off soon."
"The line was pretty bad, we got cut off a few times."
Active voice - intentional disconnection
Usually one party is responsible for the disconnection.
["cut off"] + [something]
["cut"] + [somebody/something] + ["off"]
"We were arguing on the phone and then he cut me off."
This means he hung up the phone or intentionally disconnected the line.
However you can provide some context if it was accidental.
"I think the receptionist is new, she cut off the line when she tried to transfer the call."
"Cut out" vs "Cut off"
"Cut out" means there is interference on the line.
"Cut off" means the line is broken.
"You're cutting out."
This means I cannot hear everything the other party is saying - either because there is static, or parts of words are missing.
"We were cut off."
This means the line was disconnected.
5. To suddenly interrupt someone who is speaking.
This is rude.
["cut"] + [somebody] + ["off"]
["cut off"] + [somebody]
"Will you stop cutting me off?"
"She has a bad habit of cutting off people when they're speaking."
"Cut in" vs "Cut off"
"Cut in" is just interrupting by saying something.
"Cut off" implies someone does not let you finish what you were saying.
"She cut him off while Wally was speaking."
This means she abruptly ended Wally's talk and didn't let him finish what he was saying.
"She cut in while Wally was speaking."
This means she interrupted Wally by saying something - maybe with a comment or question.
6. (informal) To abruptly overtake another vehicle with a small amount of distance.
This often happens without proper signalling and the vehicle being overtaken will often have to brake suddenly. This is very bad driving etiquette.
["cut off"] + [somebody/something]
["cut"] + [somebody/something] + ["off"]
"Cutting people off on the road are common in certain driving cultures."
"There's a six-car pile-up on the highway because some idiot tried to cut off a truck."
"Cut in" vs "Cut off"
"Cut in" is the proper phrasal verb for abruptly overtaking another vehicle within an unsafe distance.
"Cut off" implies that a vehicle blocked another vehicle's pathway.
"I just had a learner cut in (in) front of me earlier."
"I just had a learner cut me off earlier."
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