Introduction (A Funny Thing Happened on the Way to Calculus)

During the fall semester of 2005, I was slated to teach University Calculus I to a class of mostly incoming freshmen. It had been a while since I taught both the class and freshmen, so on the first day I decided to do some review and pick my students' brains. I wrote y = f(x2) on the board and asked if that was a function. The unanimous answer was yes. Without exploring that too much, I drew Figure 23.1 on the board and asked if that was a function.

The response was overwhelmingly yes and when I asked why, the response was that it passed the vertical line test. I then wrote the following on the board.

When asked if that was a function, about half of the class said it was and about half said it wasn't. From the looks on people's faces, the majority of students in the class were not sure of their answers either. After having students discuss their thoughts with their neighbors, we had a class discussion about whether it was, in fact, a function or not. The gist of people who said no was that it wasn't a function because there was no formula or graph with which to determine specific values. The people who said yes said that it was still a rule that assigned a unique value to each x.

There was still no definite consensus when I drew Figure 23.2 on the board.