Challenge for Textbooks
The undesirable convention of interchanging the variable names is deeply entrenched in the textbook culture, and has been for many decades. Moreover, it is probably
institutionalized by standardized testing. It will be a challenging task to abandon this practice and present inverse functions in a less confusing way – a task that may need to be stretched across more than one new edition of a textbook.
An intermediate version of a textbook might be envisioned as follows.
The recommended procedure for finding the inverse of a given function is:
- Write [latex]y = f(x)[/latex].
- Solve for [latex]x[/latex] in terms of [latex]y[/latex].
- Write the rule for the inverse in the form [latex]x = f^{-1}(y)[/latex].
- As an afterthought, interchange the names of the variables.
This meets a continuing (but inappropriate) expectation that [latex]x[/latex] will be retained as the independent variable. However, it should be accompanied by an explanation that the last step, while expected by tradition, is not really necessary, is detrimental for
understanding the concept of inverse functions, and should absolutely be avoided when working with any application in which the variables represent physical
quantities.
Algebra texts tend to brainwash us all into thinking that the horizontal axis must be named [latex]x[/latex] and the vertical axis [latex]y[/latex]. They should make the point that, when graphing, the independent or domain variable should be identified and its name should be placed on the horizontal axis, and its values should be plotted relative to that axis. The name of the dependent or range variable, whatever it is, should be placed on the vertical axis.
Not all college algebra texts include a discussion of inverse relations. If a textbook does present this topic, and it is recommending that the inverse be formulated by
interchanging the names of two variables governed by a defining equation, that is not constructive. Perhaps the best strategy would be just to remove the entire topic, at least temporarily. At this level, there is no subsequent application that requires a knowledge of inverse relations.
A coordinated effort will no doubt be required on the part of textbook publishers and authors and standardized testers. Nevertheless, we educators have an obligation to present this topic in the clearest way possible, so we should advocate that textbook authors who are using the traditional presentation start revising their material.
