alright just for fun: AMA but only about math, will attempt to speed-explain stuff with as few symbols and equations as possible and see what happens
(esp happy to field questions about stuff that seems basic to you and that you feel like you should've gotten a long time ago!)
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okay idk if this will address your confusion but i think there's a real important ontological point here that people elide. in mathematics functions just kinda "exist in platonic reality"; there's no expectation that a function has values that one should be able to compute
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the main thing people will say about this is "functions are not formulas," as in, you're allowed to specify their values via more complicated procedures. so a function f : X -> Y is just some "arbitrary" "assignment" of values, for each x in X, of some f(x) in Y
but this elides a serious and important point, which is, how are you allowed to specify functions then??? the formal way to do this is to describe them as subsets of the cartesian product X x Y in ZF set theory and it's not very satisfying tbh but that is the formal def
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among other things the question of which functions "exist" is sensitive to the details of set theory which personally i find very unsatisfying. but again idk if this addressed your actual question
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