The period of `f (x) =x - [x],` if it is periodic is

Options

A. f (x) is not periodic
B. `1/2`
C. 1
D. 2

Answer

To find the period of the function \( f(x) = x - [x] \), where \([x]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Understand the Function
The function \( f(x) = x - [x] \) represents the fractional part of \( x \). This means that for any real number \( x \), \( f(x) \) gives us the part of \( x \) that is after the decimal point.

### Step 2: Analyze the Fractional Part
The fractional part of \( x \) can be expressed as:
\[ f(x) = x - [x] = \{ x \} \]
where \( \{ x \} \) is the fractional part of \( x \). This value will always lie in the interval \( [0, 1) \).

### Step 3: Determine the Behavior of the Function
The function \( f(x) \) behaves as follows:
- For \( x \) in the interval \( [n, n+1) \) (where \( n \) is an integer), \( f(x) \) will increase linearly from \( 0 \) to just below \( 1 \).
- At \( x = n+1 \), \( f(x) \) will reset back to \( 0 \).

### Step 4: Identify the Periodicity
Since \( f(x) \) resets every time \( x \) increases by \( 1 \), we can conclude that the function is periodic with a period of \( 1 \).

### Step 5: Conclusion
Thus, the period of the function \( f(x) = x - [x] \) is:
\[ \text{Period} = 1 \]

### Final Answer
The period of \( f(x) = x - [x] \) is \( 1 \).

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