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Given that the composition g(f(x)) equals |sin x| and that |f(g(x))| equals (sin √x)², which of the following is consistent?

1. f(x) = sin² x, g(x) = √x
2. f(x) = sin x, g(x) = |x|
3. f(x) = x², g(x) = sin √x
4. The functions f and g cannot be uniquely determined.

Correct answer: f(x) = sin² x, g(x) = √x

Solution

The correct option is consistent because when substituting g(f(x)) with f(x) = sin² x and g(x) = √x, we get g(f(x)) = g(sin² x) = √(sin² x) = |sin x|, which matches the given condition. Additionally, |f(g(x))| with these functions results in |f(√x)| = |sin²(√x)|, which aligns with the second condition.

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