Will It Be A Legitimate Question Why Concepts Like Negative Space

Why is the domain of a function so hard for undergraduate students to understand?

[I have been teaching College Algebra and Precalculus for about a decade now.]I am afraid Alon and Shai, and several commenters on their posts, are really missing the point. Yes, formally, the domain of a function is the set of its first coördinates, where I consider a function to be a particular set of ordered pairs. But this is an advanced concept in the foundations of pure mathematics. It is not at all helpful to those first becoming familiar with the very important concept of function; these will be mostly using it in applications.In College Algebra and Precalculus courses, we say that unless specifically noted otherwise, the domain of its function is its natural domain. Formally, we define the latter as the maximal (in terms of set inclusion) set of real numbers which the expression defining the function maps to real numbers. Thus, when we say [math]f(x)=1/x[/math] without further comment, the assumed domain is [math]\mathbf{R} \backslash \{0\}[/math]. When we say [math]g(x)=\sqrt{x}[/math], the assumed domain is [math][0,\infty )[/math]. However, we do make the point that when our function is a model of a real (i.e., physical, economic, etc.) phenomenon, the domain only includes the values that make sense in the model. (For example, a price, production level or height may not be negative.) These conventions are generally followed in subsequent classes (e.g., Calculus).Of course, were we to speak to the audience to these classes of the “maximal set of real numbers in terms of inclusion…,” they would be completely befuddled. Such a perspective is way outside of this audience’s realm of discourse. I generally say, “The domain is the set of x’s where the function makes sense,” and proceed to present and get them to work on many, many examples.Now, to address the original question, viz., why do students have so much trouble with this concept, I think a big part of the problem is that students at this level have difficulty with thinking of answers to math problems being anything other than a number. Even though we may review interval notation and give many, many examples, they seem unable to deal with the concept of sets such as these. I try to address this in class. Alas, my ultimate evaluation is that most of the students who are forced to take this class have no initiative to expand their concept of mathematics. That is, they don’t want to change their view of mathematics, so they don’t.