If `f(x)=(sin([x]pi))/(x^2+x+1)` , where `[dot]` denotes the greatest integer function, then

Options

A. f is one-one
B. f is not one-one and non-constant
C. f is constant function
D. f is zero function

Explanation

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

\[ f(x) = \frac{\sin([\![x]\!] \pi)}{x^2 + x + 1} \]

where \([\![x]\!]\) denotes the greatest integer function (also known as the floor function).

### Step 1: Understanding the Greatest Integer Function
The greatest integer function \([\![x]\!]\) returns the largest integer less than or equal to \(x\). For example:
- If \(0 \leq x < 1\), then \([\![x]\!] = 0\)
- If \(1 \leq x < 2\), then \([\![x]\!] = 1\)
- If \(2 \leq x < 3\), then \([\![x]\!] = 2\)
- And so on...

### Step 2: Evaluating \(\sin([\![x]\!] \pi)\)
Since \([\![x]\!]\) is always an integer, we can evaluate \(\sin([\![x]\!] \pi)\):
- For any integer \(n\), \(\sin(n \pi) = 0\).
Thus, \(\sin([\![x]\!] \pi) = 0\) for all \(x\).

### Step 3: Simplifying \(f(x)\)
Now substituting this back into the function:
\[ f(x) = \frac{0}{x^2 + x + 1} \]
Since the numerator is zero, we have:
\[ f(x) = 0 \]

### Step 4: Analyzing the Function
The function \(f(x) = 0\) is a constant function.

### Step 5: Determining if \(f\) is One-to-One or Onto
- A constant function cannot be one-to-one (1-1) because it maps every input to the same output.
- A constant function also cannot be onto (onto) unless the codomain is restricted to the constant value itself.

### Conclusion
Thus, we conclude that \(f(x)\) is a constant function, specifically \(f(x) = 0\).

### Summary of Results
- \(f(x)\) is a constant function.
- It is neither one-to-one nor onto.