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Find the range of the function f(x) = arctan(sqrt([x] + [-x])) + sqrt(2 - |x|) + 1/x², where [*] denotes the greatest integer (floor) function.

1. [1/4, infinity)
2. {1/4} union [2, infinity)
3. {1/4, 2}
4. [1/4, 2]

Correct answer: {1/4, 2}

Solution

sqrt([x]+[-x]) is real only when x is an integer ([x]+[-x]=0). The constraint 2-|x|>=0 gives |x|<=2, and x!=0 for 1/x². So x in {-2,-1,1,2}. f(-2)=f(2)=1/4 and f(-1)=f(1)=2, giving range {1/4, 2}.

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