Thoughts on continuous functions and precalculus

Hybrid schooling is requiring a bit more use of multiple choice and other questions that can be computer graded. A closely connected issue I have been thinking about precalculus questions can be done which are calculator-impervious or calculator resistant. One recently one I came up with was of the form "Here are a bunch of graphs, which could be the graph of a rational function" (and one has then a bunch of different graphs with different discontinuities and asymptotes).

One that occurred to me was "How many discontinuities does the following function have" and then have some interesting function like f(x) = (x³-x)/((x-1)(4-|x|)).

But it occurred to me that if trig functions are allowed in the above sort of question, the problems can become non-trivial. In particular, consider the following: Set f(x) = 1/((sin pi x )² + (sin e^x )² ). Is f(x) continuous? This question is equivalent to asking if e^n /pi is ever a natural number for an integer n, since that's exactly where f(x) would be discontinuous. This is as far as I'm aware an open problem. A counterexample would in particular yield a negative solution to the very old conjecture that pi and e are algebraically independent (essentially the statement that there's no nice purely polynomial relation between the two). What I don't know: Is there some easy way to make a function f(x) which is essentially a precalculus level function where its continuity is actually equivalent to pi and e being algebraically independent.