"Check up on," "check out," and "check on" are all phrasal verbs that use the verb "check" along with a preposition or adverb. Here's a breakdown of their meanings:
"Check up on" means to investigate or monitor something or someone to make sure they are doing well or to verify their status. For example, "I'm going to check up on my elderly neighbor to see if she needs anything."
"Check out" has several different meanings, including to examine or investigate something, to leave a place, or to borrow or purchase something temporarily. For example, "I'm going to check out this new restaurant downtown," or "I need to check out of the hotel by noon."
"Check on" means to make sure that someone or something is okay or to verify the status of something. For example, "I'm going to check on my sick friend to see how they're feeling."
So, in summary, "check up on" and "check on" are similar in that they both involve verifying someone or something's status or well-being, while "check out" can refer to examining or investigating something, leaving a place, or borrowing or purchasing something temporarily.