It is a common practice to have students of elementary algebra infer the domain of a function as an exercise. I believe this is contrary to the spirit of the definition of a function as a collection of ordered pairs, no two of which have the same first term. In all serious, set-theoretic presentations of functions, relations, of which functions are a special case, come first, and then the definition of domain and range. So, if any inference is to be done, it would seem like it should be of the “formula”, or explicit rule, if there is one, that defines how the ordinate is obtained from the abscissa.
What the inference amounts to in practice is finding what values of the independent variable would result in either a denominator of 0 or the positive even root of a negative number. This sounds like a made-to-order exercise for students of elementary algebra, and so I can understand the temptation to take this path. Still, it does seem to me to go against the very spirit of the subject. Furthermore, I recall seeing someplace a discussion that ran something like this:
A. Why don’t we just agree that the domain of the function is the set of all values for which the formula is meaningful?
B. We can’t do even that, because such a maximal set of such values is not necessarily unique.
Speaker B then goes on to give an explicit example where there is more than one maximal set of numbers satisfying the formula. If I recall correctly, this diaglog was in a book on complex variables, or within a larger discussion regarding complex variables. Of course, we can easily construct a crude counterexample by citing the function that takes every number to its square root: for real numbers the domain is the set of non-negative real numbers, but for complex numbers the domain is the set of all complex numbers. However, I believe the example given by speaker B had something to do with a tricky denominator. Does anyone know the dialog/example I am referring to?
So, I believe the answer to this question is in the negative, but I wanted to see what the community thinks.
After all, here at MSE, we seem to take such exercises in stride, such as here: