give chase vs go after

give chase and go after both are verbs.

Word	Noun	Adjective	Verb	Adverb
give chase	No	No	Yes	No
go after	No	No	Yes	No

As verbs, give chase and go after are synonyms defined as:

Other synonyms of give chase include chase, chase after, dog, tag, tail, track, trail.

give chase (verb)	go after (verb)
go after with the intent to catch	go in search of or hunt for
	go after with the intent to catch