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The values of x satisfying the equation log₂(4 - x) = f(x), where f(x) = max{sqrt(|x|), |x|} for x < 0 and f(x) = min{sqrt(|x|), |x|} for x >= 0, are alpha and beta. Find [alpha] + [beta], where [.] denotes the greatest integer function.

1. -3
2. -2
3. -1
4. 0

Correct answer: -3

Solution

Careful case analysis gives two roots. For x >= 0 with 0<=x<1: log₂(4-x) = sqrt(x); testing shows x=0 gives log₂(4)=2 and sqrt(0)=0, not equal. For x>=1: log₂(4-x)=x; x=1 gives 1=1 (valid, so alpha=1). For x<0 with -1<x<0: log₂(4-x)=sqrt(-x); x approaching 0 from left gives 2 vs 0. For x<=-1: log₂(4-x)=(-x); trying x=-2: log₂(6) ~ 2.585 vs 2 (not equal); x=-2.5: log₂(6.5)~2.7 vs 2.5 (close); numerically beta ~ -2.something. [alpha]+[beta] = [1]+[-3] = 1+(-3) = -2. Re-examining: alpha=1, [alpha]=1; beta~-2.x, [beta]=-3; sum=-2. Given options, answer is -2.

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