Sharing Teaching Ideas: Understanding Functions without Using the Vertical Line Test

Abstract

Since I started teaching precalculus eighteen years ago, I have struggled with how to convey a function's definition so that its importance and usefulness comes across to students. This is especially the case with the Vertical Line Test (VLT), a mechanism for testing whether the graph of a relation is a function. (Vertical lines are drawn through the graph. If every vertical line intersects the graph in exactly one point, the relation is a function.) In this era of “learning mathematics with understanding” (NCTM 2000, p. 20), the Vertical (and Horizontal) Line Tests present a challenge. I find that once students are introduced to these devices, they tend to disregard the concepts that the devices are testing and instead mechanically apply the tests. This rote application limits not only the students' understanding of the concepts but also their ability to work within and across representations to solve problems related to identifying functions.