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Find the inverse functions. (i) Let f: [-1, 1] -> [-1, 1] be defined by f(x) = x|x|. Find f⁻¹(x). (ii) Let f(x) = 1 + e^(ln(x + 2)). Find f⁻¹(x). Which option gives both inverses correctly?

1. (i) f⁻¹(x) = sqrt(|x|) * sign(x); (ii) f⁻¹(x) = x - 3
2. (i) f⁻¹(x) = x²; (ii) f⁻¹(x) = x + 3
3. (i) f⁻¹(x) = |x|; (ii) f⁻¹(x) = ln(x) - 2
4. (i) f⁻¹(x) = -sqrt(|x|); (ii) f⁻¹(x) = e^x - 2

Correct answer: (i) f⁻¹(x) = sqrt(|x|) * sign(x); (ii) f⁻¹(x) = x - 3

Solution

(i) f(x) = x|x| equals x² for x >= 0 and -x² for x < 0, a strictly increasing odd function. For y >= 0, y = x² -> x = sqrt(y); for y < 0, y = -x² -> x = -sqrt(-y). Combined, f⁻¹(x) = sign(x) * sqrt(|x|). (ii) e^(ln(x+2)) = x + 2 (for x + 2 > 0), so f(x) = 1 + (x + 2) = x + 3. Inverting y = x + 3 gives x = y - 3, so f⁻¹(x) = x - 3.

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