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A function is defined as f(x) = (x - 2)^(1/3) for x >= 2 and f(x) = (x - 2)³ for x < 2. Which of the following correctly represents the inverse function f^(-1)(x)?

1. f^(-1)(x) = x³ + 2 for x >= 0, and f^(-1)(x) = x^(1/3) - 2 for x < 0
2. f^(-1)(x) = x^(1/3) + 2 for x > 0, and f^(-1)(x) = x³ + 2 for x <= 0
3. f^(-1)(x) = x³ + 2 for x >= 0, and f^(-1)(x) = x^(1/3) + 2 for x < 0
4. f^(-1)(x) = x³ for x >= 0, and f^(-1)(x) = x³ - 2 for x < 0

Correct answer: f^(-1)(x) = x³ + 2 for x >= 0, and f^(-1)(x) = x^(1/3) + 2 for x < 0

Solution

Swapping x and y in each branch gives x = y³ + 2 (when original output y >= 0) and x = y^(1/3) + 2 (when original output y < 0), matching option C.

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