When Can Two Functions Be Inverses in Only One Direction?

Date: 09/28/2005 at 19:28:10
From: Bhajanpreet (Bhaj)
Subject: What if one is an inverse, but the other isn't

If f and g are functions and f(g(x)) = x but g(f(x)) does NOT = x,
then what are f and g?

There are so many possibilities and I have no idea where to start; I
just keep doing trial and error with different concepts.

At first I tried doing things where f(x) = 1/x and things like that so
that sometimes the domain wouldn't allow a certain x, so the answer
would work one way, but not always sometimes, therefore the second
wouldn't be equal to x (not always at least).

Then I started working with roots and exponents, doing things like
f(x) = x^2 and g(x) = sqrt(x).  The problem with this one was that
g(f(x)) = |x| and f(g(x)) = x, but only if I put a domain on x, x>0.
But then the first part, g(f(x)) = |x|, would become g(f(x)) = x as
well because it's always positive anyways and the absolute value
doesn't matter anymore.  This was the same with every even exponent.

After that I went to odd exponents.  This didn't really work out too
well.  You probably already know why, but I tried f(x) = x^3 and
g(x) = cuberoot(x).  So, f(g(x)) = g(f(x)) = x, which doesn't really
help at all.  This goes for all odd exponents.

I also played around with absolute values, but I wasn't quite sure how
to write an inverse of that or anything close to it.  Same with floor
and roof functions.

Date: 09/28/2005 at 21:09:57
From: Doctor Vogler
Subject: Re: What if one is an inverse, but the other isn't

Hi Bhaj,

Thanks for writing to Dr. Math.  I'm glad you're trying to figure 
things out for yourself and learn new things.  This isn't an easy
question, and it doesn't have a simple answer.  I suppose that the
book means that f(g(x)) = x for all real numbers x, and g(f(x)) is
not always equal to x.  In any case, if f(g(x)) = x for all real
numbers x, then that requires that the domain of g be all real
numbers.  But what of the image of g?  (By the way, in this context
I'm using "image" in a way that you might be more familiar with as
"range".)

It turns out that if x is in the image of g, then g(f(x)) = x.  You
see, if there is some y such that x = g(y), then

  g(f(x)) = g(f(g(y))) = g(y) = x,

since f(g(y)) = y.  So the only way that we could have g(f(x)) be
something different from x is if x is not in the image of g.  So we
need the image of g to be less than all real numbers.  So we might be
tempted to try something like g(x) = x^2, but there is another problem
with this.  The function g can't take the same value at two different
places, because if

  g(a) = g(b)

then

  f(g(a)) = f(g(b))

which means that

  a = f(g(a)) = f(g(b)) = b.

So we need to find a function g whose domain is all real numbers and
whose range is *not* all real numbers.  Then we define f so that it is
the inverse of g on the image of g, and it can be anything off the
image of g.

Some examples of functions g (whose domain is all real numbers but
whose range is not) include

  g(x) = arctan(x)

  g(x) = e^x

  g(x) = (e^x - 1)/(e^x + 1)

and many, many others.  For f(x), you define f to be the inverse of g
on the range of g, and to be whatever you'd like off the range of g. 
For example, for the first function g(x) = arctan(x), you could define

  f(x) = tan(x)

and you could optionally define it on the undefined points, such as

           tan(x) if x is not pi/2 times an odd integer
  f(x) = {
           0      if x is pi/2 times an odd integer

But you really only have to define f(x) = tan(x) in the range of g,
which is the interval (-pi/2, pi/2), so you could define f as

           tan(x) if -pi/2 < x < pi/2
  f(x) = {
           0      otherwise

Either of these has f(g(x)) = x for all x, but g(f(x)) = x only for x
values in the range of g.

Does that make sense?

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

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

Date: 10/03/2005 at 17:14:49
From: Bhajanpreet (Bhaj)
Subject: What if one is an inverse, but the other isn't

Thank you so much for your extensive answer.  It really helped me
understand better.  I also found another answer to this question from
Dr. Rob (http://mathforum.org/library/drmath/view/54581.html) and
found something interesting; it was the same question, but Doctor Rob
answered by saying something about having a function on a sequence?  I
know this is a bit off the topic of my original question, but what is
a function on a sequence and how do you write that?

Thanks again!

Date: 10/03/2005 at 19:33:50
From: Doctor Vogler
Subject: Re: What if one is an inverse, but the other isn't

Hi Bhaj,

Yes, his example might be confusing if you've only dealt with
functions of real numbers.  The idea is that a function is a rule that
takes an input from one set (its domain) and gives an output from
another set (its range).  Most functions that you've seen before have
a domain that is either all real numbers or a subset of the real
numbers (such as square root has a domain of non-negative real
numbers), and whose range is also real numbers.
 
But you can define functions from any set (domain) to any set (range),
and these two sets don't even have to be related.  In Doctor Rob's
example, he is defining functions on sequences.  So f takes in a
sequence of numbers and gives out a different sequence of numbers. 
His function f drops the first term in the sequence and moves
everything else to the left by one spot.  His function g pushes
everything to the right by one spot and sticks a zero in the leftmost
spot.

The important thing to notice here is that he is still following the
rule I described.  I said:

So we need to find a function g whose domain is all real numbers and
whose range is *not* all real numbers.  Then we define f so that it is
the inverse of g on the image of g, and it can be anything off the
image of g.

I could have said, more generally,

We need to find a function g whose image is *not* all of its range.
Then we define f so that it is the inverse of g on the image of g,
and it can be anything off the image of g.

In the case of Doctor Rob's example, the range of g is the set of
sequences, but all of the values of g have the first term zero.  So f
has to be the inverse of g (shift everything left one spot) when the
input has a zero in the first term, but it can take any value when the
input has anything else in the first term (but his example has f shift
everything left one spot anyway).

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

Date: 11/29/2005 at 21:57:19
From: Bhajanpreet (Bhaj)
Subject: Thank you (What if one is an inverse, but the other isn't)

This is a REALLY late thank you--I always wanted to continue the
conversation, but I've seriously been busy.  But I still never forgot,
and I want to thank you; that was a wickedly awesome response and
really helped me understand!  Thank you, Doctor Vogler!