Finding the Inverse of y = x^x

Date: 08/31/2006 at 01:09:19
From: Eiji
Subject: Inverse function of y=x^x (x to the power of x)

Hello Dr. Math.  My name is Eiji in Grade 11, and I have recently
learned about inverse functions.  Then, I began to wonder what the
graph of y = x^x looks like.  So I downloaded software which draws the
graph automatically.

After that, I began wondering about the inverse function of y = x^x.  
The thing which I found most difficult about this is that when I swap
x and y, it gets very difficult to solve for y.

When I asked teachers, they started thinking about the natural logarithm.

y = x^x

When we swap x and y, We get

x = y^y

Taking ln from each

ln x = y ln y ...

And I got stuck.  Can you help?

Date: 08/31/2006 at 10:37:33
From: Doctor Vogler
Subject: Re: Inverse function of y=x^x (x to the power of x)

Hi Eiji,

Thanks for writing to Dr. Math.  The answer to your question is, 
unfortunately, not simple.  You have probably learned that every one-
to-one function has an inverse (those are functions that pass the 
"horizontal line test").  Consider the inverse of cosine.  Well, 
cosine is not a one-to-one function, since (for example)

  cos(360 degrees) = cos(0 degrees).

But if we restrict the cosine to only a small range of 180 degrees 
(or pi radians), then the cosine *is* one-to-one, and so it has an 
inverse.  What is the inverse?  Well, it is arccos (sometimes written 
cos^-1), which only means "the inverse of cosine."

Similarly, some functions have inverses that you can write out in a 
simple way using other elementary functions (i.e. well-known 
functions).  But most one-to-one functions have inverses that you 
can't write out using elementary functions.  Such is the case for

  f(x) = x + cos(x),

for example.

So the first question to ask is:  Is your function one-to-one?  You 
say you have graphed it.  What does the graph look like?  Does the 
function pass the horizontal line test?  (That is, does every 
horizontal line cut the graph at no more than one point?)

The answer is no.  For example,

  (1/4)^(1/4) = (1/2)^(1/2).

So the function is *not* one-to-one and therefore does not have an 
inverse.

Of course, like with the cosine function, you could restrict to a 
smaller domain on which it *is* one-to-one, and then it would have an 
inverse.  For example, if you restrict it to the domain consisting of 
x values bigger than 1/e (e the base of the natural log, 2.718...), 
then the function

  f(x) = x^x, for x > 1/e

is one-to-one, and so it has an inverse.  What is its inverse?

Well, naturally, it is f^-1(x).

Like I said, most one-to-one functions have inverses that you can't 
write out using elementary functions.  

But you can graph it!  Just take the graph of x^x (to the right of 
the lowest point at x=1/e) and reflect it over the line y=x.

Some graphing programs will even allow you to graph it as

  x = y^y

or

  ln(x) = y * ln(y).

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/