What is the range of the function f(x) = 1 / (x^2 + 4)?
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Correct answer
A. (0, 1/4]
Principle or equation
To find the range of a rational function, consider the possible values of the denominator. Since x^2 ≥ 0, x^2 + 4 ≥ 4, so 1/(x^2+4) is positive and at most 1/4. It approaches 0 as x → ±∞ but never equals 0.
Why this answer is correct
x^2 + 4 ≥ 4 for all real x. Thus 0 < 1/(x^2+4) ≤ 1/4. The maximum occurs at x=0, giving 1/4. As |x| increases, the fraction decreases toward 0 but never reaches 0. So range is (0, 1/4].
Example
For f(x) = 1/(x^2+1), range is (0,1].
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