1. I got out of my car to check the engine.
2. I got off my car to check the engine.
(Are both the same? In #2, what is the part of speech of 'off'? Is 'off' an adverb or a preposition?)
3. I got out of my car to check the engine.
4. I got out of my car to check out the engine.
5. I got out of my car to check on the engine.
(Are the three the same? What is the function of 'out' or 'on'? Are both adverbs?)
1 is correct, but 2 is not.
"Off" means you were on top of it and then you got off!
3, 4, and 5 are okay; 3 and 4 mean the same thing, but 5 is slightly different.
To check out, to check on, to get out, to get off -- all those are phrasal verbs:
http://grammar.ccc.commnet.edu/grammar/verbs.htm#phrasal
1. Yes, in both sentence 1 and 3, the meaning is the same. "I got out of my car to check the engine" means that you exited the car in order to inspect or examine the engine.
2. In sentence 2, "off" is a preposition. It is used to indicate movement away from something. "I got off my car" means that you moved away from your car, possibly by stepping down from it.
3. In sentence 3, "out" is an adverb. It describes the action of leaving the interior of the car. "I got out of my car" means that you exited the car.
4. Sentence 4 has a slightly different meaning from the previous sentences. "I got out of my car to check out the engine" means that you left the car to take a closer look at the engine or inspect it in more detail.
5. Similarly, sentence 5 has a different connotation. "I got out of my car to check on the engine" implies that you exited the car to see how the engine is doing or if it is functioning correctly. It suggests a sense of monitoring or making sure everything is okay.
In both sentences 4 and 5, "out" and "on" are adverbs. "Out" describes the action of leaving the car, and "on" describes directing attention towards the engine or checking its condition.