If `f(x)=(sin([x]pi))/(x^2+x+1)` , where `[dot]` denotes the greatest integer function, then (a) `f` is one-one (b) `f` is not one-one and non-constant (c) `f` is a constant function (d) None of these

Options

A. `f` is one-one
B. `f` is not one-one and non-constant
C. `f` is a constant function
D. None of these

Explanation

To solve the problem, we need to analyze the function \( f(x) = \frac{\sin([\![x]\!] \pi)}{x^2 + x + 1} \), where \([\![x]\!]\) denotes the greatest integer function (also known as the floor function).

### Step 1: Analyze the numerator
The numerator of the function is \( \sin([\![x]\!] \pi) \). The greatest integer function \([\![x]\!]\) returns the largest integer less than or equal to \( x \). Therefore, for any integer \( n \), we have:
\[ \sin(n \pi) = 0 \]
for all integers \( n \).

### Step 2: Determine the values of the function
Since \([\![x]\!]\) will always be an integer for any real number \( x \), we can conclude that:
\[ \sin([\![x]\!] \pi) = 0 \]
for all \( x \in \mathbb{R} \). Thus, the function simplifies to:
\[ f(x) = \frac{0}{x^2 + x + 1} = 0 \]
for all \( x \in \mathbb{R} \).

### Step 3: Check the nature of the function
Since \( f(x) = 0 \) for all real numbers \( x \), we can conclude that:
- The function is constant.
- It does not vary with \( x \).

### Step 4: Evaluate the options
Now we can evaluate the given options:
(a) \( f \) is one-one: This is incorrect because \( f(x) \) is constant (not one-one).
(b) \( f \) is not one-one and non-constant: This is incorrect because \( f(x) \) is constant.
(c) \( f \) is a constant function: This is correct since \( f(x) = 0 \) for all \( x \).
(d) None of these: This is incorrect as option (c) is valid.

Thus, the correct answer is:
**(c) \( f \) is a constant function.**