The range of the function `f(x) =[sinx+cosx]` (where `[x]` denotes the greatest integer function) is `f(x) in ` :

Options

A. [-2,1]
B. {-2,-1,0,1}
C. {-1,1}
D. {-2,0,-1}

Answer

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

### Step 1: Determine the range of \( \sin x + \cos x \)

We know that the maximum and minimum values of \( \sin x + \cos x \) can be calculated using the formula:
\[ \sin x + \cos x = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]
Thus, the range of \( \sin x + \cos x \) is:
\[ -\sqrt{2} \leq \sin x + \cos x \leq \sqrt{2} \]

### Step 2: Calculate the approximate values of \( -\sqrt{2} \) and \( \sqrt{2} \)

Calculating the numerical values:
\[ \sqrt{2} \approx 1.414 \]
So, the range of \( \sin x + \cos x \) is approximately:
\[ -1.414 \leq \sin x + \cos x \leq 1.414 \]

### Step 3: Identify the integers in the range

Now, we need to find the integers that fall within the range of \( -1.414 \) to \( 1.414 \). The integers in this range are:
\[ -2, -1, 0, 1 \]

### Step 4: Determine the greatest integer values

1. For values in the interval \( [-1.414, -1) \), the greatest integer is \( -2 \).
2. For values in the interval \( [-1, 0) \), the greatest integer is \( -1 \).
3. For values in the interval \( [0, 1) \), the greatest integer is \( 0 \).
4. For values in the interval \( [1, 1.414] \), the greatest integer is \( 1 \).

### Step 5: Compile the range of \( f(x) \)

Thus, the range of \( f(x) = [\sin x + \cos x] \) is:
\[ \{-2, -1, 0, 1\} \]

### Final Answer

The range of the function \( f(x) = [\sin x + \cos x] \) is:
\[ f(x) \in \{-2, -1, 0, 1\} \]

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