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Question 1208748: Find the domain of y = x^(x).
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52755)   (Show Source):
You can put this solution on YOUR website! .
Find the domain of y = x^(x).
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The domain of this function is the set of all positive real numbers (0,infinity).
Indeed, function  is defined for all such values of x
and is not defined for x = 0, as well as is not defined for all negative values of x.
Solved.
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comment from student: Why can't x be 0 or less than 0?
My response. Because this function is not defined there.
Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
Answer: x > 0 which is the set of positive real numbers.
The interval notation is (0, infinity)
Replace the word "infinity" with the symbol for infinity if needed.
Explanation
If x is negative, say x = -2, then we would have  which evaluates to a real number.
However, if we tried something like x = -0.5, then,
That will not result in a real number since we have a negative under a square root.
Your teacher didn't state it directly, but I'll assume that you're only working in the real number set (or I'll assume that your teacher has not covered complex numbers just yet).
To make life easy, we simply ignore the set of negative numbers entirely.
Negative integers could be allowed (since x = -2 worked just fine), but we would have an infinite collection of floating island points that couldn't be connected.
Therefore the domain is the set of positive numbers.
In terms of symbols we write x > 0
That translates to the interval notation (0, infinity)
You may be wondering: What about x = 0?
This is tricky because 0 to any nonzero exponent is 0
This is because we have a string of 0's multiplied together to result in 0.
But anything to the 0th power is 1, where the "anything" is nonzero
The right-hand-sides 0 and 1 clash together. Since they don't agree on the same value, this means there isn't full agreement on what  is
It's one of the indeterminate forms. A similar article is here
In short, we leave x = 0 out of the domain.
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