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Let f: [0, 1] -> R be a continuous function that takes only rational values. If f(0) = 2, find the value of tan⁻¹(f(1/2)) + tan⁻¹((3/2) * f(1/2)).

1. pi/4
2. pi/6
3. pi/3 + tan⁻¹(3)
4. tan⁻¹(2) + tan⁻¹(3)

Correct answer: tan⁻¹(2) + tan⁻¹(3)

Solution

The set of rational numbers Q has the property that it is totally disconnected — there are no connected subsets with more than one point. Since [0,1] is connected and f is continuous with f([0,1]) a subset of Q, the image must be connected, so it must be a single point. Hence f is constant. Given f(0) = 2, we have f(x) = 2 for all x in [0,1]. Therefore f(1/2) = 2. The expression becomes tan⁻¹(2) + tan⁻¹((3/2) * 2) = tan⁻¹(2) + tan⁻¹(3).

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