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The range of the function f(x) = (x - 1) / (p - x² + 1) does not contain any value in the interval [-1, -1/3]. Find the true set of values of p.

1. (-inf, -1)
2. (-inf, -1/4)
3. (0, +inf)
4. (-inf, 0)

Correct answer: (-inf, 0)

Solution

Setting f(x) = y and rearranging gives a quadratic in x: y*x² + x - (1 + y*(p+1)) = 0. For y not in range, discriminant < 0: 1 + 4y(1 + y(p+1)) < 0 for all y in [-1, -1/3]. This leads to the condition on p.

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