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Let f: [2, inf) -> [8, inf) be defined by f(x) = x² - (8 / (2 + a²)) * x + 2*a² + 4, where a is a real number. If f is surjective, find the product of all possible values of a.

1. sqrt(2)
2. -sqrt(2)
3. 2
4. -2

Correct answer: -2

Solution

Surjectivity requires the range of f to be exactly [8, inf). Setting f(2) = 8 or the vertex value = 8 (when vertex >= 2) gives the valid values of a, whose product is -2.

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