Inverse Functions

When we write, for example, [latex]y = ln\ x[/latex] if and only if [latex]x = e^y[/latex], it is important to stress the following:

Accordingly, when we are asked to find the inverse of a function that expresses [latex]y[/latex] in terms of [latex]x[/latex], the relevant question is:  “Can we express [latex]x[/latex] in terms of [latex]y[/latex]?”  To answer, the natural step is to solve the equation [latex]y = f(x)[/latex] for [latex]x[/latex].  If we succeed, we have an
equation of the form [latex]x = g(y)[/latex] which defines the rule for the inverse [latex]g[/latex] of the original function [latex]f[/latex].  Note that [latex]y[/latex] has become the independent variable and [latex]x[/latex] the dependent variable, and this conforms to the expectation laid out in the guiding principle stated above.  The roles played by the two variables have been interchanged, but the names of the variables have not been interchanged.

To graph an inverse function expressed in the form [latex]x = g(y)[/latex], we should identify the independent variable as [latex]y[/latex], put its name on the horizontal axis, and plot its values with reference to that axis.  The name of the dependent variable [latex]x[/latex] should go on the vertical axis and its values should be plotted against that axis.  Algebra texts have made us
uncomfortable about this.  Nevertheless, it is not appropriate to insist that the name of the independent variable must always be [latex]x[/latex].

These observations become even clearer when we are dealing with applications in which the variables represent physical quantities, especially if their names are not [latex]x[/latex] and [latex]y[/latex].  Suppose, for example, we are considering supply and demand, and [latex]p[/latex] represents the price of a commodity and [latex]q[/latex] represents the quantity traded.  If we have a model that expresses price in terms of quantity, then the inverse of that function is, very
simply, the rule that expresses quantity in terms of price.  If we were to interchange the names [latex]p[/latex] and [latex]q[/latex], then [latex]p[/latex] would represent quantity values and [latex]q[/latex] would represent price values.  This is nothing more than a source for confusion.  Interchanging the
variable names is just not a good idea.  Would anyone accept the argument that we need to exchange the names of the variables because [latex]q[/latex] should always be the name of the independent variable?  Yet textbooks typically demand that [latex]x[/latex] should be maintained as the name of the independent variable, even when it is not appropriate.

See also [1], an article entitled Inverse Functions:  We’re Teaching It All Wrong!