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To Invert Functions, First Subvert Routine

Date: 12/09/2010 at 22:47:28
From: Maureen
Subject: inverse function notation

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 the x's and
             y = 3x + 7         y's and then solve for y -- but
                                Wikipedia doesn't

     (y - 7)/3 = x              From my point of view this is 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

Date: 12/10/2010 at 23:12:29
From: Doctor Peterson
Subject: Re: inverse function notation

Hi, Maureen.

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

(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.

To make the result look more like you are used to, just note that the
variable in a function definition is a place-holder; you can use any name.
Their answer gives f^-1(y); to find f^-1(x), just change the name of the
variable to x, and you get

   f^-1(x) = (x - 7)/3
In making that replacement, you have in essence finally swapped the
variables by putting them into their traditional roles, though that isn't
necessary in order to define the function.

It's really important to distinguish the letter from the role, and I think
they got it exactly right by focusing on the meaning rather than what
letter is being used. As you said, what they produce IS the inverse, using
y as the variable. The variable doesn't matter; the function does!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 

Date: 12/11/2010 at 09:24:06
From: Maureen
Subject: inverse function notation

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.

Date: 12/11/2010 at 23:28:15
From: Doctor Peterson
Subject: Re: inverse function notation

Hi, Maureen.

I considered last time using the log and exponential functions as an
example, but chose not to in case you were not familiar with them. Let's
use the natural log to take what I said last time, and make it a little
more concrete -- after some review.

Your difficulty is in equating the usual roles of x and y with the
explicit statement of a function. It is perfectly valid to use different
letters, like "A = g(t)," in defining a function. Since that is true, it
has to be valid also to use x and y in places different from the usual way
students first see; it is quite common to see "x = f(t)" with x the
dependent variable; and it is equally valid to write "x = h(y)." The roles
the variables play are indicated not by the letters used, but by the way
they are used in the function statement.

Similarly, if I write an equation that shows how to calculate one variable
from another (what I call a formula), that explicitly expresses a
function, though without naming it. Writing ...

   y = 2x + 1 

... expresses y as a function of x. Writing ...

   x = (y - 1)/2

expresses x as a function of y. 

This is a DIFFERENT function (namely the inverse of the first) because now
x is the dependent variable, not the independent variable. (This doesn't
change the relation between x and y, however; the inverse function is
really the same relation viewed from a different perspective.)

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

   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.

As for graphing, this gets us back to the traditional roles of x and y,
where the independent variable is called x and is on the horizontal axis.
There is nothing sacred about that; you can put any variable on either
axis, as in my example of x depending on t, where you would put t on the
horizontal axis and x on the vertical. You could graph an inverse function
by actually swapping the axes without renaming them:

  ^            o
  |           o
  |          o
  |        o
  |     o

This graph represents the exponential function because the dependent
variable on the vertical axis is the exponential of the independent
variable on the horizontal axis. Of course, having the axes labeled this
way is awkward, so we would more naturally swap the names of the variables
when we graph the function, always keeping in mind that the names are just
placeholders and do not affect what the function IS.

Again, yes, all this can confuse students, especially if they have been
treated the usual way, shielded from the fact that variable names do not
matter, and almost always given x as the independent variable. But they
will be far less confused later on if they can learn this lesson early and
understand what a function IS: a relationship that is independent of what
you happen to name the variables involved. The approach to inverses that
is troubling you is the more mature approach, and students who are going
to go on in math will need to understand it eventually; so why make it
hard for them later by trying to make things "easy" now and getting them
used to an immature viewpoint?

Just recently, I borrowed an exam from a colleague to use in a course that
was new to me, and I noticed that a question was worded this way:

   "Determine algebraically whether the given equation 
   represents a function."
I corrected that in my version to 

   "Determine algebraically whether the given equation
   represents y as a function of x."

It's a common mistake to assume that all functions take x as the
independent variable -- but it is a mistake, and I don't want my students
to make it.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Functions

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