hello

say i have two functions, and and i wanted to find where they intersect. since is a line, if i could rotate both functions so that became the x-axis, , then to find the intersection i would solve the now rotated by setting , then un-rotating the solutions.

firstly, how can i accomplish this rotation transform, and secondly, can i do more complex transforms, like and transformed so that becomes the x-axis? again, i would also have to de-transform the solutions. if i was at all fluent in calculus (took 4 semesters worth of it, haha, just don't remember much), then i'm sure i'd have a better idea on how to figure this out, or at least ask better questions.

*edit*

for this one, i visualize flattening out one curve, and the other one obeys the same movements. mathematically i think it would be taking lines perpindicular to and rotating each one into as in the first case.

and for the real kicker (or maybe you'll just kick me), how about and ? ..and again, de-transformation.

*edit*

transforming certain functions would cause the other to no longer be a function..not a good idea haha.

but,

given the range could these last functions be modeled by a polynomial?

thanks for any help or ideas!