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/
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