2019 AMC 10B Problems/Problem 9
Contents
Problem
The function is defined by for all real numbers , where denotes the greatest integer less than or equal to the real number . What is the range of ?
Solution 1
There are four cases we need to consider here.
Case 1: is a positive integer. Without loss of generality, assume . Then .
Case 2: is a positive fraction. Without loss of generality, assume . Then .
Case 3: is a negative integer. Without loss of generality, assume . Then .
Case 4: is a negative fraction. Without loss of generality, assume . Then .
Thus the range of the function is .
~IronicNinja
Solution 2
It is easily verified that when is an integer, is zero. We therefore need only to consider the case when is not an integer.
When is positive, , so
When is negative, let be composed of integer part and fractional part (both ):
Thus, the range of x is .
Note: One could solve the case of as a negative non-integer in this way:
Solution 3 (Formal)
Let {} denote the fractional part of ; for example, {}, and {}. Then for , {} and for , {}.
Now we can rewrite , breaking the expression up based on whether or .
For , the above expression is equal to {} {}{}
.
For , the expression is equal to {} {}
{}.
Therefore the only two possible values for , and thus the range of the function, is .
~KingRavi
Solution 4
We have 2 cases: either is positive or is negative.
Case 1 - x is positive:
Let , where is a positive integer and is a positive real number between 0 and 1. We have and , so the possible value of if is positive is .
Case 2 - x is negative:
Let , where is a positive integer and is a positive real number between 0 and 1. We have and
and , so the possible values of if is negative are and
Hence, the possible values of are and , so the answer is . ~azc1027
Video Solution
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
2019 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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All AMC 10 Problems and Solutions |
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