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Find the greatest interval contained in (−π/2, π/2) on which the function f(x) = 4^(−x²) + cos⁻¹((x/2) − 1) + log(cos x) is defined.

1. [−π/4, π/2)
2. [0, π/2)
3. [0, π]
4. (−π/2, π/2)

Correct answer: [0, π/2)

Solution

The function f(x) is defined where each component is valid: 4^(-x²) is defined for all x, cos⁻¹((x/2) - 1) requires (x/2) - 1 to be in [0, 1], which restricts x to [0, 2], and log(cos x) is defined where cos x > 0, limiting x to (−π/2, π/2). The intersection of these conditions gives the interval [0, π/2).

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