If the domain of the function `f(x)=([x])/(1+x^(2))`,where `[x]` is greatest integer `lex,` is `[2,6],` then its range is

Options

A. `((5)/(37),(2)/(5))`
B. `((5)/(26),(2)/(5))]-{(9)/(29),(27)/(109),(18)/(89),(9)/(53)}`
C. `((5)/(26),(2)/(5))]`
D. `((5)/(37),(2)/(5))-{(9)/(29),(27)/(109),(8)/(89),(9)/(53)}`

Answer

To find the range of the function \( f(x) = \frac{[x]}{1 + x^2} \) where the domain is given as \( [2, 6) \), we will follow these steps:

### Step 1: Identify the intervals for the greatest integer function
The greatest integer function \( [x] \) takes on constant values over intervals. For the domain \( [2, 6) \), we can break it down into sub-intervals based on the values of \( [x] \):
- For \( 2 \leq x < 3 \), \( [x] = 2 \)
- For \( 3 \leq x < 4 \), \( [x] = 3 \)
- For \( 4 \leq x < 5 \), \( [x] = 4 \)
- For \( 5 \leq x < 6 \), \( [x] = 5 \)

### Step 2: Write the function in each interval
Now we can express \( f(x) \) in each of these intervals:
- For \( 2 \leq x < 3 \):
\[ f(x) = \frac{2}{1 + x^2} \]
- For \( 3 \leq x < 4 \):
\[ f(x) = \frac{3}{1 + x^2} \]
- For \( 4 \leq x < 5 \):
\[ f(x) = \frac{4}{1 + x^2} \]
- For \( 5 \leq x < 6 \):
\[ f(x) = \frac{5}{1 + x^2} \]

### Step 3: Find the maximum and minimum values in each interval
Next, we will find the maximum and minimum values of \( f(x) \) in each interval.

1. **For \( 2 \leq x < 3 \)**:
\[ f(2) = \frac{2}{1 + 2^2} = \frac{2}{5} = 0.4 \]
As \( x \) approaches 3, \( f(x) \) approaches:
\[ f(3) = \frac{2}{10} = 0.2 \]
So, the range in this interval is \( [0.2, 0.4] \).

2. **For \( 3 \leq x < 4 \)**:
\[ f(3) = \frac{3}{10} = 0.3 \]
As \( x \) approaches 4:
\[ f(4) = \frac{3}{17} \approx 0.176 \]
So, the range in this interval is \( [0.176, 0.3] \).

3. **For \( 4 \leq x < 5 \)**:
\[ f(4) = \frac{4}{17} \approx 0.235 \]
As \( x \) approaches 5:
\[ f(5) = \frac{4}{26} \approx 0.154 \]
So, the range in this interval is \( [0.154, 0.235] \).

4. **For \( 5 \leq x < 6 \)**:
\[ f(5) = \frac{5}{26} \approx 0.192 \]
As \( x \) approaches 6:
\[ f(6) = \frac{5}{37} \approx 0.135 \]
So, the range in this interval is \( [0.135, 0.192] \).

### Step 4: Combine the ranges
Now, we need to combine the ranges from all intervals:
- From \( [2, 3) \): \( [0.2, 0.4] \)
- From \( [3, 4) \): \( [0.176, 0.3] \)
- From \( [4, 5) \): \( [0.154, 0.235] \)
- From \( [5, 6) \): \( [0.135, 0.192] \)

### Step 5: Determine the overall range
The overall range is the union of all these intervals:
- The minimum value is \( \frac{5}{37} \) (approximately \( 0.135 \)).
- The maximum value is \( \frac{2}{5} \) (approximately \( 0.4 \)).

Thus, the range of the function \( f(x) \) is:
\[ \left[ \frac{5}{37}, \frac{2}{5} \right] \]

### Final Answer
The range of the function \( f(x) = \frac{[x]}{1 + x^2} \) where \( x \in [2, 6) \) is:
\[ \left[ \frac{5}{37}, \frac{2}{5} \right] \]