Technically the Shuttle was a reusable rocket, so we know it works.
Reusable conventional rockets are an interesting concept. Engineers at NASA and ESA thought it would not be economical. NASA and ESA use a small number of large engines each time they send one of their rocket up, and so reusing them means firing one of the large engine for a long time (which damages them and requires significant amounts of fuel that would have been otherwise used to put satellite into orbit). A SpaceX rocket uses a large number of small engines, so letting one fire for longer in order to retrieve all the others makes more sense. Nonetheless these engines spend more time in space and experience a reentry, which may damage them.
AFAIK SpaceX hasn't yet tried to reuse any of the retrieved engine. They already have a less-than-stellar safety record so that is not surprising. I suspect the road to effective return on investment on the reusable rocket venture may be longer than anticipated.