Two functions with the same domain and region but defined by different rules, will be distinguished by different function names. However, for the function which is 0 for all rational x and 1, otherwise, there is no formula that uses only broadly accepted math notations. Of course, if a function is used very often mathematicians may decide to standardize its name. After that point on such a named function may be used as a legitimate part of a formula. The word association is not often used as a substitute for a function perhaps because it's judged to be more vague or fundamental than function .
Transformation is the term used in geometry, mapping appears in topology.
This way a function is identified with its graph . Alexander Bogomolny 2789865 .