How to Find an Inverse Function: Conflicting Approaches

While pondering issues that often give students trouble in algebra, I decided to check what we have said in Ask Dr. Math about inverse functions. I discovered four answers (all, as it happens, written by me – I tend to be attracted to certain topics!) to essentially the same question, spread over 13 years. It is one that I have pondered in teaching: having taught the subject from books that use two different approaches to the task, I find that I prefer the one that seems to be used less commonly, but each of them has its benefits. Let’s take a look at one of these four discussions, then quickly look at the others to supplement it. In 2010, Maureen wrote this:

To Invert Functions, First Subvert Routine

The inverse of a function is found by interchanging x's and y's, right? However, on Wikipedia they determine the inverse in a way that I find confusing. Specifically, I am writing what they do on the left and my confusion on the right.

       f(x) = 3x + 7

                         Normally, I would now switch
          y = 3x + 7     the x's and y's and then solve
                         for y -- but Wikipedia doesn't

                         From my point of view this is
  (y - 7)/3 = x          NOT the inverse -- it is the
                         original function

f(inv) of y = (y - 7)/3  This is the inverse using y as
                         the variable

Most books do not do it this way; and although I agree with the final answer, I find it somewhat meaningless. Would you agree?

I am including the usual way of finding the inverse.

       f(x) = 3x + 7
          y = 3x + 7     given the original function
          x = 3y + 7     switch x and y
          y = (x - 7)/3  solve for y to get inverse function

My reply was that, in fact, I prefer the method used by Wikipedia here, which I imagine is less often used in current textbooks (at least at an introductory level).  There are a couple reasons to prefer it. One is that only this way is appropriate to applications in which the variables, unlike the typical generic x and y, have specific meanings, so that they can’t be interchanged (see below for an example); another is that this method forces us to get out of the rut of assuming that x is always the independent variable, and y is always dependent.

What they're doing is correct, and in fact is what I prefer. The confusion is probably because you are used to always thinking of y as a function of x.

(It troubles me that texts often ask questions like "Does the equation x + y^2 = 1 represent a function?" when they really mean to ask if it represents y as a function of x. In that example, x is a function of y, though y is not a function of x. A function is about the relationship between two variables, not what they are called.)

What Wikipedia has done is not to exchange the NAMES of the variables in the function, as usual, but just to change their ROLES. By solving for x, they are determining how x (the "input" of f) can be found given y (the "output" of f). That is exactly what it means to find an inverse.

If you recognize that x and y are just dummy variables, you see that Wikipedia’s form,  \(f^{-1}(y) = \frac{y \, – \, 7}{3}\), and Maureen’s form, \(f^{-1}(x) = \frac{x \, – \, 7}{3}\), are really the same function, just with different names for the variable. By not changing the variable names at the start, Wikipedia ends up with y as the independent variable – not what most students are used to, but perfectly  legal. And in fact, the main idea of an inverse function is precisely that we are changing the roles of the variables: what was the input of the original function becomes the output of the inverse function, and vice versa. What they are called is far less important that what they mean.

But Maureen needed more help:

Maybe my difficulty is from a graphing point of view.

If you graph ...

   y = 3x + 1 

... and you create a table of values for x and y, and then you graph ...

   x = (y - 1)/3

... you get the same graph, since the tables are the same. When graphing an inverse from a table of values, you specifically interchange the x's and y's, because if you do not, you have the same table.

I feel students (or possibly just me) confuse a different way of writing a function with the inverse function. Specifically y = ln x is the same function as e^y = x. One is NOT the inverse of the other. The inverse of y = ln x is x = ln y, or e^x = y.

The two equations she graphed are actually  different functions (y as a function of x, and x as a function of y); but they are equivalent equations (relating the same pairs of x and y), which is why they are represented by the same graph. Maureen has confused functions with equations. In reality, the second function, \(f^{-1}(y) = \frac{y \, – \, 1}{3}\), is the inverse of the first, \(f(x) = 3x + 1\), where x and y have kept the same name but changed roles.

Now consider the logarithm. This is written explicitly as a function the name of which is "ln" rather than "f" or "g," but it is the same idea. If I write ...

   y = ln(x)
   
... I am using the ln function to express y as a function of x. This equation also expresses (implicitly) x as a function of y, since ln is one-to-one. When you solve for x to make the latter function explicit, you have

   x = e^y

This explicitly states a different function than ...

   y = ln(x)
   
... although the relation between the variables is the same.

If we were to name the new function, calling it exp, so that ...

   x = exp(y) 
     = e^y
   
... the named function exp clearly is not the same function as ln. But nothing has changed except for giving it a name. The equation expresses x as a function of y, named or not.

So when we interchange the variables and change y = ln(x) to x = ln(y), we are, as you say, inverting the function -- in the sense of what function y is of x. We have changed the relationship between x and y. But in another sense, it is still the same function (as explicitly written), just expressed with different placeholders.

So there are two perspectives on what constitutes an inverse. In one sense, just swapping the variables implicitly inverts the relationship, by swapping implied roles (x becomes y); but the inverse is shown explicitly by solving for the other variable, regardless of what you call it (input becomes output).

My answer to another of the four questions I referred to above includes an example of the situation where swapping of variables is meaningless, and you must use the other approach:

Inverting, Subverted

Both approaches are valid. The first is best when the variable names mean something, so that changing their names would not make sense. For example, you wouldn't say that the inverse of C = f(r) = 2 pi r is f^-1(r) = r/(2 pi), where r has now become the circumference. Rather, you would say that r = f^-1(C) = C/(2 pi).

Your trouble, I believe, is that you are thinking of the variable names as if they had a fixed meaning somewhat like that, but using the method of inverting that requires you to swap names. You ask about "the inverse ofy" rather than of the FUNCTION, making y a name for the function itself. You can't invert a variable, only a function. And if you do think of y as the same thing as f(x), you can't swap variable names and still say that.

In fact, in my circumference example above, I was initially going to call the function C(r), which is done commonly, naming the function for its output; but it would make no sense to call the inverse C^-1. If anything, we would call the inverse r(C), since it gives the value of r that yields a given circumference C. I didn't do that because it would not fit the inverse notation you are using, and would just add confusion. (If it does, ignore this paragraph!)

I prefer the first method of inverting I showed, because it helps to free students from fixating on x as the independent variable, and avoids the change of meaning that is confusing to you and others. Unfortunately, I have to teach the second in algebra classes, because it is what most students seem to see elsewhere.

For the record, the other two questions I referred to are

Graphs of Inverse Functions
Inverting Functions

The final thanks in the latter, a long, rambling discussion, are worth ending with:

The terminology and explanation has solidified my understanding. I believe my confusion was, as you pointed out, that "An equation does not define a function!" This is very important and I have never thought of it that way. It could be because I never read that somewhere or was never taught that or most importantly never had to think of them that way until now.