Domain and Range of x^x

Date: 08/24/98 at 21:19:37
From: Larry
Subject: Domain and range

What is the domain and range of x^x?

Date: 08/24/98 at 23:15:02
From: Doctor Jaffee
Subject: Re: Domain and range

Hi Larry,

The value of x can't be 0, because 0^0 is undefined. Consequently, y 
can't be 0 either because the only exponential number that equals 0 
has 0 for a base.

If x is a positive number, it can be shown that the smallest value of 
y occurs when x = 1/e, where e is approximately 2.71. So, x can be any 
positive number, but y must be greater than (1/e)^(1/e).

Now, when x is negative, it gets really complicated. If x is a negative 
odd integer, then y will be negative, but between 0 and -1.  If x is a 
negative even integer, then y will be positive between 0 and 1. If x is 
a negative rational number between 0 and -1, which, when written as a 
fraction has an even numerator, then y will be positive and greater 
than 1.  

I've given you a start on this problem. What you need to think about 
to finish it is how much bigger than 1 can y get in the previous case, 
what are all the other cases, and what happens then.

I know I've left the hardest part of the problem for you, but give it 
a try and see what you come up with.  Write back if you have more 
questions about this problem or when you have questions about other 
math problems. 

- Doctor Jaffee, The Math Forum
Check out our web site! http://mathforum.org/dr.math/