CTK Insights

I cannot recollect which book it was but I do remember it was a book by a famous mathematician who wrote in Introduction that to write a book is the best way to learn a new topic. I got this memory flash when reading an article in Mathematics Teacher:

Our teachers misled us, but we don't blame them. They were only teaching what was in the textbook. And as new teachers—because of our lack of experience and our reliance on the textbook—we continued to teach the procedure we had learned as students. It wasn't until we began writing textbooks ourselves that we were compelled to confront the inverse function falsehoods in our intellectual past. These contradictions were difficult to detect because they were broadly accepted and perpetuated in widely used textbooks.

The online version of the magazine (Mathematics Teacher Volume 104, No. 7 March 2011) also links to several articles from previous years that addressed the same problem of inverse functions from very different perspectives.

Let $f$ be a function with domain $A$ and range (or codomain) $B$. If $f$ is 1-1 (injective) function, it has an inverse, denoted $f^{-1}$ which is a function characterized by two properties:

1. The domain of $f$ is the range of $f^{-1}$ and the range of $f$ is the domain of $f^{-1}$.
2. $f$ and $f^{-1}$ undo each other's actions: $f^{-1}(f(x)) = x$ for $x\in A$ and $f(f^{-1}(x)) = x$ for $x\in B$.

This definition, although correct, is already a source of confusion to many students. The problem is that the students are accustomed to functional notions $y = f(x)$ and terminology that makes a distinction between $x$ as independent and $y$ as depedent variables. So students would expect the second identity to be written as $f(f^{-1}(y)) = y$ for $y\in B$. This is quite natural: if $y = f(x)$ then surely $x = f^{-1}(y)$.

This is of course true but conflicts with the tendency of having the independent variable run over the horizontal axis, the dependent one of the vertical axis. Thus students have to learn the mysterious fact that the graphs of the inverse functions are reflections of each other in the diagonal $y = x$.

The authors of the article I quoted from at the beginning see a solution to the problem in never drawing the graphs of the inverse functions in the same system of coordinates. For example, if $C$ and $F$ denote the temperature in the Celsius and Fahrenheit scales then, as well known, $F = f(C) = 1.8C + 32$. Of course, $C = f^{-1}(F) = (F - 32)/1.8$. The graph of the former is drawn with the horizontal $C$-axis, that of the latter with the horizontal $K$-axis. Other examples include using $\theta$ to denote the argument of trigonometric functions, like y = sin(\theta), and, in general, Keeping track of the meaning of the variables is considered critical. As the authors observe,

Although the equations $y = f(x)$ and $y = f^{-1}(x)$ may be graphed simultaneously on the same set of axes, the physical interpretation of the context being modeled is lost.

And later

The horizontal axis cannot simultaneously represent both x and y, so graphing both functions on the same set of axes is impossible.

Candidly, I am under impression that the authors may need to write more books to master the subject. Not only it is possible to graph both functions on the same set of axes, it takes no special effort to do so. In fact the same set of points serves as the graph of both functions in the same system of coordinates. The graphs of $y = f(x)$ and $y = f^{-1}(x)$ are exactly the same set of points in the same coordinate axes. The only effort required to recognize this fact lies in shifting the view point of which of the variables is "dependent" and which is "independent".

The most serious misconception related to graphing inverse functions is the strong belief that the independent variable must be always associated with the horizontal axis. This precept leads to difficulties in handling coordinate transformations, implicit functions, and three dimensional coordinates - to name a few topics of a more advanced character.

The authors of a 2007 article lament a general misunderstanding among preservice teachers of the connections between different inverses. On a midterm exam the authors posed the following question:

Following is a representative student's response:

The meanings of the "–" in the expressions have nothing to do with each other. In $?3$ it means "negative" or "opposite" of three. Since $3$ is on the right side of $0$, $?3$ is on the left side of $0$. In the expression $4^{?3}$ it means to write the $4^3$ in the denominator instead of the numerator. It's kind of like using the reciprocal. In $tan^{-1}(3)$ it could mean to put the tan in the denominator, but usually it means to take the arctan $3$. $f^{-1}(3)$ just means to take the inverse function. Mathematicians use the negative sign to mean many different things, but it is usually obvious what the teachers want by the context or what you have been studying recently.

A 1997 article skipped methodological discussions in favor of a practical advice as to how help students familiarize with the concept of the inverse function. It's a game. The teacher thinks of a number which students have to discover. How? They form a function (in the article with four operations). The teacher gives the value of the function for the secret number. Student have to run the operations in reverse. For example, four students suggested four steps to construct a function: Determine the logarithm of $n$, add $7$, divide by $2$, square the result. Combined, the four steps led to $f(n) = \big(\frac{\log (n) + 7}{2}\big)^2.$ The author reports great enthusiasm among her students and eventual realization that the inverse function undoes what a function did.

Very different articles. Very different concerns. And probably very different effects.

References

1. F. C. Wilson, S. Adamson, T. Cox, A. O'Bryan, Inverse Functions: What Our Teachers Didn't Tell Us, Mathematics Teacher Volume 104, No. 7 (March 2011) 501-507
2. P. J. Maida, Can You Guess My Number?, Mathematics Teacher No. 90 (February 1997) 114–17
3. C. C. Benson and M. Buerman, The Inverse Name Game, Mathematics Teacher Volume 101, No. 2 ( September 2007) 108 – 112

1. Then as now ...