If [x] means the greatest integer less than or equal to x and {x} means the least greater than or equal to x and |x| means the absclute value value of x, then ` {x}` equal to :

Options

A. a. `-1`
B. b. `0`
C. c. `1`
D. d. any one of (a),(b),(c)

Answer

To solve the problem, we need to find the value of the fractional function \({x}\) in terms of the greatest integer function \([x]\). The fractional function is defined as:

\[ {x} = x - [x] \]

Where \([x]\) is the greatest integer less than or equal to \(x\).

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- The greatest integer function \([x]\) gives the largest integer that is less than or equal to \(x\).
- The fractional function \({x}\) gives the difference between \(x\) and the greatest integer less than or equal to \(x\).

2. **Using the Formula**:
- We can express the fractional part of \(x\) as:
\[ {x} = x - [x] \]

3. **Analyzing Different Cases**:
- **Case 1**: If \(x\) is an integer (e.g., \(x = 3\)):
- Then \([x] = 3\).
- So, \({x} = 3 - 3 = 0\).
- **Case 2**: If \(x\) is a positive non-integer (e.g., \(x = 2.5\)):
- Then \([x] = 2\).
- So, \({x} = 2.5 - 2 = 0.5\).
- **Case 3**: If \(x\) is a negative non-integer (e.g., \(x = -2.5\)):
- Then \([x] = -3\) (since -3 is the greatest integer less than -2.5).
- So, \({x} = -2.5 - (-3) = -2.5 + 3 = 0.5\).

4. **Generalizing the Result**:
- From the above cases, we can conclude that:
- For any real number \(x\):
- If \(x\) is an integer, \({x} = 0\).
- If \(x\) is a non-integer, \({x}\) will always be in the range \(0 < {x} < 1\) for positive \(x\) and \(0 < {x} < 1\) for negative \(x\).

5. **Conclusion**:
- Therefore, the fractional part \({x}\) can be expressed as:
\[ 0 \leq {x} < 1 \]

### Final Answer:
The fractional function \({x}\) is defined as:
\[ {x} = x - [x] \]
And it satisfies:
\[ 0 \leq {x} < 1 \]