The number of integers belonging to the interval [−3,30] but not belonging to the range of the function f(x)=x{x}−x[−x],x∈R, is (where [.] and {.} denote the greatest integer function and fractional part respectively.
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Solution
Consider only integers in this case
f(x)=x{x}−x[−x]=x(x−[x])−x[−x]=x2−x([x]+[−x])=x2
x∈[−3,30]⟹f(x)∈[0,900]
The integers belonging to the range are 0,1,4,9,16,25
Number of integers does not belonging to range are 34−6=28