Functions that are their own inverses

Date: 02/28/98 at 23:33:53
From: Mike
Subject: Inverse functions

Other than f(x) = x, can a function ever be its own inverse?

Date: 03/01/98 at 01:39:24
From: Doctor Mike
Subject: Re: Inverse functions

Hello,

It's nice to hear from another Mike, and another non-traditionalist as 
well.
   
The idea of an inverse function is an important one, so a function 
which is its own inverse is pretty special. Here are a few standard 
examples (besides the one example you gave) of the inverse function:

     N(x) = -x     for every number x

     R(x) = 1/x    for every non-zero number x

You can determine that these 2 functions equal their inverses by 
verifying that N(N(x))=x and R(R(x))=x for every number x that makes 
sense for the function.

Here's how that would go for R, which is the Reciprocal function.  
Take x=5, for example.  R(5)=1/5 , so 
   
     R(R(5))  =  R(1/5) = 1/(1/5) = 5.
  
For the function N, N(N(5)) = N(-5) = -(-5) = 5.

Here is a slightly more complicated function which is its own inverse:  
E(x) = 2 - x, which can also be written as E(x) = -x + 2, which are 
equivalent.  You can see that E(E(x))=x for every x:
   
     E(E(x))  =  E(-x + 2)
              =  -( -x + 2 ) + 2
              =  -(-x) - 2 + 2  
              =  -(-x) + 0 
              =  x.

I have just shown in a mechanical way that if you start out with any 
number x, then "do" the E function on it, and then "do" the E function 
on that result, you always get back to where you started.  It might 
help for you to see this in a physical setting, not just as 
manipulations of symbols and numbers.

Imagine that you have a tall metal ladder standing by a deep well, and 
there is a rope-ladder hanging down into the well. Think about the 
following three instructions, which tell you what to do in all 
possible situations.
   
     1. If you are on the metal ladder, X feet above ground
        level,
          (a) Climb down to ground level,
          (b) Climb down the rope ladder into the
              well to a Depth of X feet, and then
          (c) Climb upwards 2 feet.
     2. If you are on the rope ladder, X feet below ground
        level,
          (a) Climb up to ground level,
          (b) Climb up the metal ladder to a Height 
              of X feet, and then
          (c) Climb upwards 2 feet.
     3. If you are at ground level already, climb up 2 feet.

This covers all possibilities of what your elevation is at the start.  
No matter where you start, if you carry out these instructions twice, 
you will wind up at exactly the same place you started.
   
For example, suppose you start at 7 feet above ground. The first 
instruction is the one that applies, so you climb down from the 
ladder, climb down to 7 feet below ground, and then climb back up 2 
feet, leaving you at 5 feet below ground.  Now we follow the 
instructions a second time. You are below ground, so instruction 2 is 
the one that applies. It tells you to climb up to ground level, go up 
the metal ladder to a height of 5 feet above ground, and then to climb
up 2 more feet, leaving you at 7 feet above ground. You are back where 
you started.  This always works, whether you start at a positive 
elevation (above ground, i.e., on the metal ladder), or at a negative 
elevation (below ground, i.e., on the rope ladder), or at a zero 
elevation (at exactly ground level).

Why?  Because the function is its own inverse. The connection between 
the function and the example is that if X represents the elevation 
above(+) or below(-) ground, E(x) gives the new elevation after 
following the instructions.
  
To put this example back in terms of symbols: if we start at 7 feet 
above ground, then X=7. Since E(7)=-5 and E(-5)=7, E(E(7)) = E(-5) = 
7, which is where you began.

This should give you something to think about. Hope it helps.

-Doctor Mike,  The Math Forum
 http://mathforum.org/dr.math/