What is the range of the function f(x) = 1 / (x^2 + 2x + 3), where x is a real number?
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Correct answer
A. (0, 1/2]
Principle or equation
To find the range of a rational function with a quadratic denominator, determine the minimum or maximum value of the denominator and consider the reciprocal. The denominator must be positive for all real x if its discriminant is negative and leading coefficient positive.
Why this answer is correct
Let d(x) = x^2 + 2x + 3 = (x+1)^2 + 2. Since (x+1)^2 ≥ 0, d(x) ≥ 2, with minimum value 2 at x = -1. As x → ±∞, d(x) → ∞, so f(x) → 0. Therefore, f(x) = 1/d(x) takes values from just above 0 up to 1/2, inclusive of 1/2. Thus range is (0, 1/2].
Example
For f(x) = 1/(x^2 + 1), denominator ≥ 1, so range is (0, 1].
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