How do you find the input given the output?

Given a function, how do you compute what that function is taking in as input?

I don't see how to answer these questions. They seem to be the same question, but at different levels.

The fact that the functions that people are discussing are not known should not be very surprising. Indeed, there are many functions that are not known (or even can be proven not to exist). So in that sense, all these functions are equally mysterious. The fact that the functions are known only for the specific domains they are discussed (for example polynomials) has to do with how their definitions are given.

What is the input output of a function?

, like so?

If you want to write code that works the same in every language and does the same in each, the function should return whatever object or value the language requires for the function to be applied. An int function would return an int, so that it could be applied by another int function (in this case you'd use apply(x, x) as opposed to apply(x)). If you think of a function as a black box which takes a number and returns a number or a string or a bool or whatever, then the answer to your question is "input, output". The exact definition of what that means varies based on language, but if you consider only the operations that a function can perform, the term makes sense.

Eg if a function receives some input value and returns a string, then it's a function that accepts a string as input and produces a string as output. If a function accepts one argument and produces one return value, it is defined as input-output.