Period of `f(x) = sgn([x] +[-x])`is equal to (where [.] denotes greatest integer function

Options

A. 1
B. 2
C. 3
D. does not exist

Explanation

To find the period of the function \( f(x) = \text{sgn}([x] + [-x]) \), where \([.]\) denotes the greatest integer function, we will analyze the function step by step.

### Step 1: Understanding the Greatest Integer Function
The greatest integer function \([x]\) returns the largest integer less than or equal to \(x\). For example:
- \([4.3] = 4\)
- \([-4.3] = -5\)

### Step 2: Analyzing the Function
We can rewrite the function \( f(x) \):
\[ f(x) = \text{sgn}([x] + [-x]) \]
Here, \([-x]\) is the greatest integer less than or equal to \(-x\).

### Step 3: Evaluating the Function for Non-Integer Values
Let’s evaluate \(f(x)\) for a non-integer value, say \(x = 4.3\):
- \([4.3] = 4\)
- \([-4.3] = -5\)

So,
\[ [x] + [-x] = 4 + (-5) = -1 \]
Thus,
\[ f(4.3) = \text{sgn}(-1) = -1 \]

### Step 4: Evaluating the Function for Integer Values
Now, let’s evaluate \(f(x)\) for an integer value, say \(x = 3\):
- \([3] = 3\)
- \([-3] = -3\)

So,
\[ [x] + [-x] = 3 + (-3) = 0 \]
Thus,
\[ f(3) = \text{sgn}(0) = 0 \]

### Step 5: Generalizing the Function
From the above evaluations, we can see:
- For integer values \(x\), \(f(x) = 0\).
- For non-integer values \(x\), \(f(x) = -1\).

### Step 6: Graphing the Function
The graph of \(f(x)\) will show:
- A value of \(0\) at every integer point.
- A value of \(-1\) for all other real numbers.

### Step 7: Determining the Period
Since the function \(f(x)\) takes the value of \(-1\) for all non-integer values and \(0\) at integer values, it repeats its behavior every unit interval. Therefore, the period of the function is:
\[ \text{Period} = 1 \]

### Final Answer
The period of \(f(x) = \text{sgn}([x] + [-x])\) is equal to \(1\).

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