I might draw some ire for this one. I'm not much of a pure mathematician. I don't care much for making elegant proofs, the properties of oddball trig functions, or generalizing the sequence for making inscribed polygons (although they are pretty).
Clearly, because I don't care should mean nothing to the people that enjoy that sort of thing. Go nuts, ignore me.
Where these highly abstract tasks go awry is when you try to put them in your high school textbook. Is this an implication that high school students should be capable of these? Am I bad teacher for skipping them? What are these problems trying to accomplish?
There are many offenders, but rational functions, you by far can be the worst.
Granted, these were towards the end of a rather lengthy problem set. But what are these then just a way to find a problem that will take 10 minutes or so to simplify? Assuming the student doesn't trip over the many many algebra hurdles lurking in here.
The last one is interested in the roots of a random rational monster.
Nasty asymptotes abound, and there appears to be only one solution. I could see a more advanced math class tackling this function. Certainly I made one of you reach for a pencil to see if you could find the answer numerically (honestly, I almost did too). But why is this in high school math?
At least the headings are up front about it. My college Calc I professor had us tackle something like the first ones. I have zero interest in messing with that bottom one.
These kind of exercises have a place, but I don't think it's in high school. We just aren't given the time to make the foundational skills solid enough. My main criticism of curriculums that are a mile long is that there is no time to stop and really spend the hours working towards mastery of a skill. If your academic Algebra II sections are anything like mine, all the algebra hurdles from the book exercises would completely obscure whatever I was trying to teach about a rational function. It's a problem in Pre-Cal and Calculus. Strong algebra skills get sacrificed in the name of these random special cases because it's on the semester sheet.
Anecdotally, rational functions are big targets for the "when will we use this in life" crowd, in which case I agree with the frustrated student.