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...

• A. \(0\)
• B. \(1\)
• C. \(2\{x\}\)
• D. \(x\)

Hint:

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.

Answer 1

The Correct Option is B

Step 1: Test with a number
Take $x=2.3$: $\{x\}=0.3$ and $\{-2.3\}=-2.3-(-3)=0.7$.

Step 2: Add
$0.3+0.7=1$, which matches the second option and confirms the general result.

Step 3: Match the option
The sum 1 is the second option.

Final Answer:
The sum $\{x\}+\{-x\}$ equals 1 for non-integer x. \[ \boxed{\text{(B) }1} \]