If [x] denotes the greatest integer less than or equal to ` x and n in N , ` then ` f(X)= nx+n-[nx+n]+tan""(pix)/(2)` , is

Options

A. a periodic function with period 1
B. a periodic function with period 4 .
C. not periodic
D. a periodic function with period 2.

Explanation

To solve the problem, we need to analyze the function given:

\[ f(x) = nx + n - [nx + n] + \tan\left(\frac{\pi x}{2}\right) \]

where \([x]\) denotes the greatest integer less than or equal to \(x\) and \(n\) is a natural number.

### Step 1: Simplifying the Function
First, we can rewrite the function by recognizing that the term \(nx + n - [nx + n]\) represents the fractional part of \(nx + n\). The fractional part of a number \(y\) is defined as:

\[ \{y\} = y - [y] \]

Thus, we can express \(f(x)\) as:

\[ f(x) = \{nx + n\} + \tan\left(\frac{\pi x}{2}\right) \]

### Step 2: Analyzing the Periodicity of Each Component
1. **Periodicity of the Fractional Part**:
The fractional part function \(\{y\}\) has a periodicity of 1. Therefore, the fractional part \(\{nx + n\}\) has a periodicity of:

\[ \text{Period of } \{nx + n\} = \frac{1}{n} \]

2. **Periodicity of the Tangent Function**:
The function \(\tan\left(\frac{\pi x}{2}\right)\) is periodic with period \(2\). This is because the tangent function has a period of \(\pi\), and scaling the input by \(\frac{\pi}{2}\) gives a period of:

\[ \text{Period of } \tan\left(\frac{\pi x}{2}\right) = 2 \]

### Step 3: Finding the Overall Periodicity
To find the overall periodicity of the function \(f(x)\), we need to find the least common multiple (LCM) of the individual periods:

- Period of \(\{nx + n\} = \frac{1}{n}\)
- Period of \(\tan\left(\frac{\pi x}{2}\right) = 2\)

To find the LCM of \(\frac{1}{n}\) and \(2\):

1. Convert \(2\) into a fraction with the same denominator:

\[ 2 = \frac{2n}{n} \]

2. Now we have two fractions: \(\frac{1}{n}\) and \(\frac{2n}{n}\). The LCM of the numerators \(1\) and \(2n\) is \(2n\), and the common denominator is \(n\).

Thus, the overall LCM is:

\[ \text{LCM}\left(\frac{1}{n}, 2\right) = \frac{2n}{n} = 2 \]

### Conclusion
The periodicity of the function \(f(x)\) is \(2\).

### Final Answer
The periodicity of \(f(x)\) is \(2\).

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