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If f(x) = x|x|, then its inverse function f⁻¹(x) is equal to

1. sqrt(|x|)
2. (sgn x) * sqrt(|x|)
3. -sqrt(|x|)
4. Does not exist

Correct answer: (sgn x) * sqrt(|x|)

Solution

f(x) = x|x| equals x² for x >= 0 and -x² for x < 0; it is strictly increasing, hence invertible. Solving y = x|x| gives x = sgn(y)*sqrt(|y|), so f⁻¹(x) = sgn(x)*sqrt(|x|).

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