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If f: [1, inf) -> [1, inf) is defined by f(x) = 2^(x(x-1)), find f⁻¹(x).

1. (1/2)^(x(x-1))
2. (1/2)*(1 + sqrt(1 + 4*log₂(x)))
3. (1/2)*(1 - sqrt(1 + 4*log₂(x)))
4. Not defined

Correct answer: (1/2)*(1 + sqrt(1 + 4*log₂(x)))

Solution

y = 2^(x² - x) => log₂(y) = x² - x => x² - x - log₂(y) = 0. Quadratic root: x = (1 +/- sqrt(1 + 4*log₂(y)))/2. Since domain is x >= 1 we take the + root. Replacing y by x: f⁻¹(x) = (1/2)*(1 + sqrt(1 + 4*log₂(x))).

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