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Let f(x) = arcsin(x + 2) + arccos((x² + x + 1) / (-x)), and let g be the inverse function of f. Which of the following statements is/are correct? (Here sgn(x) denotes the signum function.)

1. sgn(f(g(x))) = 1
2. sgn(f(g(x))) = -1
3. g(pi/2) = -1
4. f(x) is discontinuous at x = -1

Correct answer: g(pi/2) = -1

Solution

The domain of f is the single point x = -1, where f(-1) = arcsin(1) + arccos(1) = pi/2 + 0 = pi/2. Since g is the inverse, g(pi/2) = -1. Also f(g(x)) = x, so sgn(f(g(x))) = sgn(pi/2) = 1, making option A also true.

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