The range of the function `f(x)=2+x-[x-3]` is, (where `[.]` denotes greatest integer function):

Options

A. `[5,6]`
B. `[5,6)`
C. `R`
D. `I`

Explanation

To find the range of the function \( f(x) = 2 + x - [x - 3] \), where \([.]\) denotes the greatest integer function, we can follow these steps:

### Step 1: Rewrite the function
We start by rewriting the function:
\[ f(x) = 2 + x - [x - 3] \]
We can express \([x - 3]\) as \([x] - 3\) since the greatest integer function shifts the value down to the nearest integer.

### Step 2: Simplify the function
This gives us:
\[ f(x) = 2 + x - ([x] - 3) = 2 + x - [x] + 3 = x - [x] + 5 \]

### Step 3: Identify the fractional part
The term \(x - [x]\) represents the fractional part of \(x\), denoted as \(\{x\}\). The fractional part \(\{x\}\) is always in the range:
\[ 0 \leq \{x\} < 1 \]

### Step 4: Substitute the fractional part into the function
Now we can rewrite \(f(x)\):
\[ f(x) = \{x\} + 5 \]

### Step 5: Determine the range of \(f(x)\)
Since \(\{x\}\) varies from 0 to just below 1, we can find the range of \(f(x)\):
\[ f(x) = \{x\} + 5 \implies 5 \leq f(x) < 6 \]

### Step 6: Conclusion
Thus, the range of the function \(f(x)\) is:
\[ (5, 6) \]