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Find the range of f(x) = (x² + x + 2)/(x² + x + 1).

1. [1, 2]
2. [1, inf)
3. [2, 7/3]
4. (1, 7/3]

Correct answer: (1, 7/3]

Solution

f(x) = 1 + 1/(x² + x + 1). Let g = x² + x + 1 = (x + 1/2)² + 3/4, so g in [3/4, inf). Then 1/g in (0, 4/3]. Hence f = 1 + 1/g in (1, 1 + 4/3] = (1, 7/3]. The value 1 is not attained (would require g -> inf).

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