2019 AMC 10B Problems/Problem 9
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
{
}
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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