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Determine the bounds (range) of the function f(x) = x² / (x⁴ + 1) for real x.

1. 0 <= f(x) <= 1/2
2. 0 <= f(x) <= 1
3. -1/2 <= f(x) <= 1/2
4. 0 < f(x) < 1

Correct answer: 0 <= f(x) <= 1/2

Solution

f(0) = 0. For x != 0, f = 1/(x² + 1/x²). By AM-GM x² + 1/x² >= 2 (equality at x = +-1), so f <= 1/2. Also f >= 0 always (numerator and denominator positive), with f = 0 only at x = 0. Hence 0 <= f(x) <= 1/2.

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