Frank Swenton - Middlebury College
The function is well known as one of the most fundamental concepts in all of mathematics, and it is equally well known as one of the most misunderstood by students in mathematics courses. Some would say that this is simply the nature of the student, and others would blame the advent of the graphing calculator. But the root cause of this problem lies much deeper, its seeds planted more than 300 years ago: it is the graph itself, which through the primacy it has attained as a means of dealing with functions, has obscured—and even supplanted—in the minds of students the very concept it was meant to illuminate.
In the Beginning, there were the Domain and the Range, each a distinct set in its own right, and there was the Function sending each element of the domain to one element of the range. Domain was domain, range was range, and never the twain did meet. Fast-forward to the 17th century: René Descartes rotates the range by 90 degrees and clanks it down right on top of the domain; upon these axes he draws a function’s graph: a construction both revolutionary and visually seductive—in fact, enough so to eclipse the function concept itself.
Considering functions via their graphs is a practice so deeply ingrained in the mathematics classroom that it escapes the critical eye given to all other aspects of mathematical pedagogy. Graphs readily provide easy answers (in some cases), but the answers are all too ready, too easy, and—most importantly—quite often incomplete or deceptively facile. X has become synonymous with “input,” and y with “output”; the derivative means “slope”; the integral means “area”; continuity means that a graph can be drawn “without lifting the pencil.” Even the very heart of being a function has sadly devolved into the “vertical line test.”
The fundamental problem is that these rough and ready geometric answers often serve to distract the learner from well-warranted further consideration of the true nature of each concept; they provide just enough of a simple, soothing “answer” with which to be content and to quiet any call for further thought. Moreover, when a learner is presented time and time again with graphs, it becomes far too easy to see only the graphs. When applying a function f to a value, the eyes quickly move directly to the point of the graph at the correct horizontal location, and there they stop. Seldom does the eye ever stray to the domain, at which the action starts, or the range, at which the action ends—the function’s meaning as the link from one to the other is lost, all attention drawn by the immense visual pull of its graph.
The effects of this graph-centric perspective on the function are felt far beyond the topic of the function concept itself. Inverse functions have students immediately flipping the plane or wrenching their necks; that inverses are simply rules sending elements of the range back whence they came becomes secondary. Differential approximation becomes the circuitous unraveling of a triangle tangent to a graph, finally arriving at a formula that is a direct consequence of the definition of derivative. The epsilon-delta definitions of limits and continuity become a crisscrossed clutter of horizontal and vertical lines on a graph, yielding a similar tangle in the learner’s mind, when the domain and range have no business intersecting in the first place. Careful inspection across the gamut of function-related concepts shows that graphs very often complicate or confound in their effort to simplify and explain.
Is the graph logically inconsistent? No. But we must reply equally in the negative to the question of whether a graph properly expresses the function concept in its entirety. One might rebut that the modern definition of the function from X to Y is as a particular subset of the Cartesian product X x Y. However, consider the logical definition of the implication “if P, then Q”: do we properly conceptualize it as “Q or not P,” or is this merely the logical statement that defines its formal meaning? A function, while encoded as a subset of X x Y, is conceptually no more a subset of X x Y than a poem is ink on a page; the difference is that no one forgets that a poem must be read aloud (at least in the mind) to be fully appreciated or understood.
A function acts: it sends each element of its domain to some element of its range; it maps subsets of the domain to their images in the range; it even pulls back sets in the range to their preimages in the domain. A function acts—and a graph simply sits, dead. We must view that static graph as a machine ready to act, or else the graph only serves to gloss over the concept; we must make the crucial effort not to be content with the limited perspective on functions afforded us by graphs alone. The graph is only a valid tool for studying a function when viewed not as the function itself, but as a representation of the function—only in vigilance of this crucial distinction will the concept retain its integrity; and with a healthy independence of graphs, the function concept can properly grow and flourish within the minds of students.