What is the range of the function f(x) = 3 / (x^2 + 1)?
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Correct answer
A. (0, 3]
Principle or equation
For a rational function with a positive denominator, the range is determined by the minimum and maximum values of the denominator and the behavior as x approaches infinity.
Why this answer is correct
The denominator x^2 + 1 is always at least 1 (minimum at x=0) and grows without bound as |x| increases. Therefore, f(x) = 3 / (x^2 + 1) has maximum value 3/1 = 3 at x=0. As x → ±∞, the denominator → ∞, so f(x) → 0 from above. Since the denominator is always positive, f(x) is always positive. Thus the range is (0, 3].
Example
For f(x) = 2 / (x^2 + 4), the maximum is 2/4 = 0.5 at x=0, and as x→∞, f→0, so range is (0, 0.5].
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