Problem:
The function f is defined by
f(x)=⌊∣x∣⌋−∣⌊x⌋∣
for all real numbers x, where ⌊r⌋ denotes the greatest integer less than or equal to the real number r. What is the range of f?
Answer Choices:
A. {−1,0}
B. the set of nonpositive integers
C. {−1,0,1}
D. {0}
E. the set of nonnegative integers
Solution:
If x≥0, then ∣x∣=x, so ⌊∣x∣⌋=⌊x⌋. Furthermore, if x≥0, then ⌊x⌋≥0, so ⌊⌊x⌋∣=⌊x⌋. Therefore f(x)=⌊x⌋−⌊x⌋=0 when x≥0.
Otherwise, x<0, so ∣x∣=−x.
If x<0 and x is an integer, then ⌊∣x∣⌋=⌊−x⌋=−x and ∣⌊x⌋∣= ∣x∣=−x. Therefore f(x)=(−x)−(−x)=0 in this case.
If x<0 and x is not an integer, then ⌊∣x∣⌋=⌊−x⌋=−⌊x⌋−1 and ∣⌊x⌋∣=−⌊x⌋. Therefore f(x)=(−⌊x⌋−1)−(−⌊x⌋)=−1 in this case.
Thus the range of f(x) is (A){−1,0}.
The problems on this page are the property of the MAA's American Mathematics Competitions