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Let f(x) = x - [x] , x in R then f(1/2)...

Let f(x) = `x - [x] , x in R ` then `f(1/2) `is

A

1

B

0

C

`-1`

D

none of these

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The correct Answer is:
To solve the problem, we need to evaluate the function \( f(x) = x - [x] \) at \( x = \frac{1}{2} \), where \( [x] \) denotes the greatest integer less than or equal to \( x \). ### Step-by-step Solution: 1. **Identify the function**: The function is given as: \[ f(x) = x - [x] \] 2. **Substitute \( x = \frac{1}{2} \)**: We need to find \( f\left(\frac{1}{2}\right) \): \[ f\left(\frac{1}{2}\right) = \frac{1}{2} - \left[\frac{1}{2}\right] \] 3. **Calculate the greatest integer \( \left[\frac{1}{2}\right] \)**: The greatest integer function \( [x] \) gives the largest integer less than or equal to \( x \). For \( \frac{1}{2} \): \[ \left[\frac{1}{2}\right] = 0 \] This is because 0 is the largest integer that is less than or equal to \( \frac{1}{2} \). 4. **Substitute back into the function**: Now, substitute \( \left[\frac{1}{2}\right] \) back into the equation: \[ f\left(\frac{1}{2}\right) = \frac{1}{2} - 0 \] 5. **Simplify the expression**: Thus, we have: \[ f\left(\frac{1}{2}\right) = \frac{1}{2} \] ### Final Answer: \[ f\left(\frac{1}{2}\right) = \frac{1}{2} \]
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