If you're just trying to explain what f~O(g) means, then it makes sense to talk in terms of functions and asymptotes like you would in an intro to calculus, sur

If you're just trying to explain what f~O(g) means, then it makes sense to talk in terms of functions and asymptotes like you would in an intro to calculus, sure. Then, separately, computer scientists are interested in f(n) = max_{|S|=n} steps_required(program, S) where S is some input/problem type with some notion of size. The fact that f looks innocuous but is secretly actually this complicated thing is perhaps the stumbling block for people. The actual O(*) part is just "line stays at or below other line once you go far enough out" (with a slight tweak to allow you to pre-stretch the bounding line).