TITLE: Is a simplified function the same as the original?
QUESTION [5 upvotes]: Is a simplified function the same as the original?
Example: 
Let $f(x) = \frac{ax}{x}$, and $g(x) = a$
where $a$ and $x$ are real numbers.
Does $g$ = $f$?

REPLY [2 votes]: In contemporary mathematics, a function should always be defined by means of: (1) a set called "domain of definition"; (2) a set called co-domain; (3) a "rule" or a "law" that sends every element of the set (1) to one (and only one) element of the set (2). If you do not like to think in terms of subsets of cartesian products, a function is a triple. As you have been told, calculus is abundant of functions as they were used by pioneers: just rules.
This can be tolerated at very low levels of teaching, but should be replaced by a more precise definition rather soon. And we, teachers and professors, should stop asking our student "to find the domain of definition of the following function", since there is no function without a domain.