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Find the range of the function f(x) = log₂( log_(1/2)( x² + 4x + 4)).

1. (-infinity, infinity)
2. (-infinity, 0)
3. [0, infinity)
4. (0, infinity)

Correct answer: (-infinity, infinity)

Solution

The inner expression is (x+2)² which ranges over (0, infinity) excluding 0. For the outer log₂ to be defined, log_(1/2)((x+2)²) must be positive, requiring (x+2)² in (0,1). On that interval log_(1/2) ranges over (0, infinity), and log₂ of (0, infinity) is all real numbers, so the range is all reals.

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