If `f(x)=(x^(3)+x+1)tan(pi[x])` (where, `[x]` represents the greatest integer part of x), then

Options

A. domain of f(x) is `R-(2n+1)(pi)/(2),n in I`
B. range of `f(x) in R`
C. `f(x)` is an even function
D. `f(x)` is a non - periodic function

Answer

To solve the problem, we need to analyze the function \( f(x) = (x^3 + x + 1) \tan(\pi [x]) \), where \([x]\) denotes the greatest integer part of \(x\).

### Step 1: Understand the function components
The function consists of two parts:
1. \( g(x) = x^3 + x + 1 \) - a polynomial function.
2. \( h(x) = \tan(\pi [x]) \) - a trigonometric function based on the greatest integer function.

### Step 2: Analyze the greatest integer function
The greatest integer function \([x]\) takes any real number \(x\) and gives the largest integer less than or equal to \(x\). For example:
- If \( x = 2.3 \), then \([x] = 2\).
- If \( x = 3.7 \), then \([x] = 3\).

### Step 3: Analyze the behavior of \(\tan(\pi [x])\)
The tangent function \(\tan(\pi n)\) (where \(n\) is an integer) is equal to 0 for all integer values. Thus, for any \(x\) such that \([x] = n\), we have:
\[ \tan(\pi [x]) = \tan(\pi n) = 0 \]
This means that the function \(h(x)\) will always be 0 for any value of \(x\).

### Step 4: Determine \(f(x)\)
Since \(\tan(\pi [x]) = 0\) for all \(x\), we can substitute this into the function:
\[ f(x) = (x^3 + x + 1) \cdot 0 = 0 \]
Thus, \(f(x) = 0\) for all \(x\).

### Step 5: Determine the domain and range
- **Domain**: Since \(f(x)\) is defined for all real numbers, the domain is \(\mathbb{R}\).
- **Range**: The function is constantly 0, so the range is \{0\}.

### Step 6: Determine if the function is even or odd
A function \(f(x)\) is even if \(f(-x) = f(x)\) for all \(x\). Since \(f(x) = 0\) for all \(x\), we have:
\[ f(-x) = 0 = f(x) \]
Thus, the function is even.

### Step 7: Determine if the function is periodic
A function is periodic if there exists a positive number \(T\) such that \(f(x + T) = f(x)\) for all \(x\). Since \(f(x) = 0\) for all \(x\), it is trivially periodic with any \(T > 0\).

### Summary of Results
- **Domain**: \(\mathbb{R}\)
- **Range**: \{0\}
- **Even or Odd**: Even
- **Periodic or Non-Periodic**: Periodic