It is a multi-word verb combining a verb and an adverb (adverbial particle). Together the two words usually have a different meaning from what the individual meanings of the words suggest (e.g, blow up). Verb + preposition combinations (e.g., look at) are sometimes called phrasal verbs (there is disagreement about this). Phrasal verbs can be transitive or intransitive.
Transitive phrasal verbs may be separable or inseparable, e.g., take off is separable: take off your hat/take your hat off.
'Look after' is inseparable: I looked after the children. NOT *I looked the children after.