What is the range of the function f(x) = 1 / (x^2 + 2x + 5), where x is a real number?
Show the answer and explanation
Correct answer
A. (0, 1/4]
Principle or equation
The range of a rational function can be found by determining the possible values of the denominator. For a quadratic denominator that is always positive, the range is (0, 1/minimum].
Why this answer is correct
Complete the square: x^2 + 2x + 5 = (x+1)^2 + 4. The minimum value of the denominator is 4, so the maximum of f is 1/4. As x → ±∞, denominator → ∞, so f → 0. Since denominator is always positive, f > 0. Thus range is (0, 1/4].
Example
For f(x) = 1/(x^2 + 1), denominator min = 1, so range is (0,1].
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