Sure - but that's just as true in reverse.
> Understanding rational functions helps to understand what vertical asymptotes are and as such are a fundamental source of examples when learning limits. They also aid in understanding why tan(x) has vertical asymptotes where cos is 0. Every complete algebra curriculum includes rational functions.
Meh. x^-1 is a good example of some things, sure, but I don't remember ever doing addition of rational functions which is what you originally talked about, and I went through an extremely reputable maths degree.
No middle school, and I very much doubt it. Searching I can see them mentioned in a further maths GCSE (which is something most schools including the one I went to don't offer, and rather suggests they're not on the regular maths GCSE, which would match my memory).
> Most students who know that x + 5/3 x is 8/3 x have trouble, initially, with understanding that 2xy+ay is (2x + a)y.
Who know that first or who have been taught it? I genuinely would like to see any actual evidence that the latter is objectively more difficult than the former.
Only anecdotal evidence. I’ve taught beginning algebra courses at a community college for 23 years. Students who don’t know fractions have a very hard time in algebra. Those who can’t understand that x + 5/3 x is 8/3 x have a hard time understanding that 2xy+ay is (2x + a)y.
Understanding rational functions helps to understand what vertical asymptotes are and as such are a fundamental source of examples when learning limits. They also aid in understanding why tan(x) has vertical asymptotes where cos is 0. Every complete algebra curriculum includes rational functions. I say complete because algebra is usually broken up into 3 courses (2 at the pre-college level).