Question:
If \( f(x) = \lfloor x \rfloor + \lfloor -x \rfloor \), where \( \lfloor \cdot \rfloor \) is the greatest integer function, then the value of \( f(2.5) \) is:
If \( f(x) = \lfloor x \rfloor + \lfloor -x \rfloor \), where \( \lfloor \cdot \rfloor \) is the greatest integer function, then the value of \( f(2.5) \) is:
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Key Fact: \( \lfloor x \rfloor + \lfloor -x \rfloor = -1 \) when \( x \notin \mathbb{Z} \)
Updated On: May 27, 2025
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The Correct Option is B
Solution and Explanation
To find \( f(2.5) \) where \( f(x) = \lfloor x \rfloor + \lfloor -x \rfloor \), let's evaluate each part of the function:
Step 1: Calculate \( \lfloor 2.5 \rfloor \)
The floor function \( \lfloor x \rfloor \) returns the greatest integer less than or equal to \( x \). Thus, \( \lfloor 2.5 \rfloor = 2 \).
Step 2: Calculate \( \lfloor -2.5 \rfloor \)
For \( \lfloor -x \rfloor \), we apply the floor function to \( -2.5 \). Since \( -2.5 \) is between \(-3\) and \(-2\), the greatest integer less than or equal to \(-2.5\) is \(-3\). Hence, \( \lfloor -2.5 \rfloor = -3 \).
Step 3: Compute \( f(2.5) \)
Substitute the results from the previous steps into the function:
\( f(2.5) = \lfloor 2.5 \rfloor + \lfloor -2.5 \rfloor = 2 + (-3) = -1 \).
Therefore, the value of \( f(2.5) \) is \(-1\).
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