2003 AMC 12A Problems/Problem 25
Let . For how many real values of is there at least one positive value of for which the domain of and the range of are the same set?
The function has a codomain of all non-negative numbers, or . Since the domain and the range of are the same, it follows that the domain of also satisfies .
The function has two zeroes at , which must be part of the domain. Since the domain and the range are the same set, it follows that is in the codomain of , or . This implies that one (but not both) of is non-positive. If is positive, then , which implies that a negative number falls in the domain of , contradiction. Thus must be non-positive, is non-negative, and the domain of the function occurs when , or
Completing the square, by the Trivial Inequality (remember that ). Since is continuous and assumes this maximal value at , it follows that the range of is
As the domain and the range are the same, we have that (we can divide through by since it is given that is positive). Hence , which both we can verify work, and the answer is .
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