Would speakers rated at 80-200 watts function with a integrated amp rated at 50 watts… thanks

Yes they would function.

It depends a lot on your speakers sensitivity, size of room and how loud you play your music…

While I am not an expert…my .02 feelings is that your amp is under powered for speakers rated at a
minimum of 80 watts… you may well be on your way to blowng your tweeters should your amp clip
while trying to feed your speakers a stronger than normal transient signals…

Consider upgrading your amp…speakers will really appreciate it

Others with more knowledge willno doubt step in to advise even more

Hope this helps

Why don’t you give more info. Like what speakers? What integrated? What size room would it be in and how loud do you play?