We can consider the fraction 1/2 to be a function with the domain of even natural numbers. At this point we don’t do signs. So the numbers are pure natural numbers. We consider these a type of size.
If we think of natural numbers as in factored form, then what the function 1/2 does is remove one of the factors of 2. The function has domain limited to the even natural numbers, so the are either 0 or have a factor 2 in them.
So 1/2 as ordered pairs is (0,0), (2,1),(4,2),(8,4), etc.
The input followed by the output.
The function 1/3 has as domain the numbers with a 3 in their factorization and we include 0 as well. The graph of a function is the set of ordered pairs. So the graph of the function 1/3 is
(0,0), (3,1),(6,2),(9,3), etc.
An operator is simply a function defined on some set of test data. So a fraction as a function is the same as a fraction as an operator. Here same is same, not just equal.