This video explains evidence of the history of various types of archery technology.  It is not known when the first bow and arrow was invented, but the oldest definitive evidence of archery comes from late Paleolithic Europe (about 17 thousand years ago) in the form of bows, arrows, arrowheads, and cave art.  It is possible that “composite” bows have been implemented in prehistoric times (as opposed to being made out of only one solid piece of wood.)  Archery was not used for hunting mammoths because they had mostly died out by the time archery was widely adopted.

[Note: I think archery is an impressive technology for back then, since it requires the combination of many innovations.  Archery would have been more difficult to develop than pottery that essentially only requires setting a clay object into a fire to harden it, yet pots weren’t implemented in Mesopotamia until relatively recent at 6,000 years ago!  Why the delay?  (However pots have been found in China that date to 29-12 thousand years ago.)]