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Graphs of Inverse Functions

Date: 07/25/2002 at 16:57:07
From: Jonida
Subject: Graphs of inverse functions

Suppose y=f(x) and y=g(x) are inverses. It turns out that the graph 
of one function is the graph of the other, but reflected across the 
line y=x. Explain why this is the case.


Date: 07/26/2002 at 12:43:16
From: Doctor Peterson
Subject: Re: Graphs of inverse functions

Hi, Jonida.

There are several ways to think about this. I'll suggest a couple.

First, let's take a graph:

    y             y=f(x)
    |            /
    |           /
    |        /
    |     /
    |   /
    |  /
    +-------------x

The inverse function means that

    y = f(x) whenever x = g(y)

and

    x = g(y) whenever y = f(x)

That means that the point (x,y) is on the graph of y = f(x) whenever 
the point (x,y) is on the graph of x = g(y). That is, the graph we 
have drawn is also the graph of x = g(y):

    y             x=g(y)
    |            /
    |           /
    |        /
    |     /
    |   /
    |  /
    +-------------x

But that's not the graph we want; we want to see the graph of y=g(x). 
How do we get that? We have to swap the variable names:

    x             y=g(x)
    |            /
    |           /
    |        /
    |     /
    |   /
    |  /
    +-------------y

Now we have the right variables, but the axes are in the wrong places.

Let's make these two graphs, for y=f(x) and y=g(x). If you've drawn 
the second graph on sufficiently transparent paper, you can just flip 
the paper over and place it on top of the first, so that the x and y 
axes on the two graphs line up together. Essentially what you have 
done is to rotate the paper around the line y=x:

    y             y=x       x             y=x
    |           /           |           /
    |         /             |         /
    |       /       ----->  |       /
    |     /                 |     /
    |   /                   |   /
    | /                     | /
    +-------------x         +-------------y

That's the same as reflecting every point on the paper in that line, 
so our graph of g is now the reflection of the graph of f we started 
with.

Here's a less visual and more analytical way to see it. We'll start 
the same way:

We make the graph:

    y             y=f(x)
    |            /
    |           /
    |        /
    |     /
    |   /
    |  /
    +-------------x

The inverse function means that

    y = f(x) whenever x = g(y)

and

    y = g(x) whenever x = f(y)

That means that the point (a,b) is on the graph of y = f(x) whenever 
the point (b,a) is on the graph of y = g(x), since b=f(a) means the 
same thing as a=g(b). So we can look at these two points:

    y
    |   Q(b,a)  /
   a+---*     /
    |   |   /
    |   | /
   b+---/-----*P(a,b)
    | / |     |
    +---+-----+---x
        b     a

Now, what does it mean to say that Q(b,a) is the reflection of P(a,b) 
in the line y=x? One way you can think of that is that the line 
segment PQ between the points is (1) perpendicular to y=x, and (2) 
bisected by y=x. Can you see that? You might want to imagine folding 
the paper along that line, so that P and Q coincide, and think about 
what the line PQ will look like.

Now to show that PQ is perpendicular to y=x, we need to show that its 
slope is -1. That's true, right?

And to show that the midpoint of PQ is on y=x, we need to find the 
midpoint and show that its coordinates are equal. You can do that 
too, right? Then you've proved it!

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

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Functions

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