Question:
If \(f(x)=\{x\}\) represents the fractional part function, then for any \(x\notin \mathbb{Z}\), the expression \(f(x)+f(-x)\) is equal to...
If \(f(x)=\{x\}\) represents the fractional part function, then for any \(x\notin \mathbb{Z}\), the expression \(f(x)+f(-x)\) is equal to...
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Write the fractional part as x minus the greatest integer, and use that the greatest integer of -x is minus the greatest integer of x minus 1 for non-integers.
Updated On: Nov 12, 2026
- \(0\)
- \(1\)
- \(2\{x\}\)
- \(x\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
\(\{x\}=x-[x]\), where \([x]\) is the greatest integer function.
Step 2: Calculation
For \(x\notin\mathbb{Z}\), \([-x]=-[x]-1\).
\(\{-x\}=-x-[-x]=-x+[x]+1\)
\(\{x\}+\{-x\}=x-[x]-x+[x]+1=1\)
Step 3: Compare the options
The sum is 1, the second option.
Final Answer:
The expression equals 1. \[ \boxed{\text{(B) }1} \]
\(\{x\}=x-[x]\), where \([x]\) is the greatest integer function.
Step 2: Calculation
For \(x\notin\mathbb{Z}\), \([-x]=-[x]-1\).
\(\{-x\}=-x-[-x]=-x+[x]+1\)
\(\{x\}+\{-x\}=x-[x]-x+[x]+1=1\)
Step 3: Compare the options
The sum is 1, the second option.
Final Answer:
The expression equals 1. \[ \boxed{\text{(B) }1} \]
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