consider a function f(x)= [sin x] +[cos x] where [.] represents greatest integer function, range of f(x) is?

We are given the function:

\[
f(x) = \lfloor \sin x \rfloor + \lfloor \cos x \rfloor
\]

where \(\lfloor . \rfloor\) represents the greatest integer function (or floor function). To determine the range of \(f(x)\), we need to analyze the behavior of \(\lfloor \sin x \rfloor\) and \(\lfloor \cos x \rfloor\), since the sine and cosine functions take values between \(-1\) and \(1\).

### Step 1: Analyze the possible values of \(\lfloor \sin x \rfloor\)
- The sine function \(\sin x\) takes values in the interval \([-1, 1]\) for all \(x\).
- Therefore, \(\lfloor \sin x \rfloor\) will take values from the set \(\{-1, 0\}\), because:
  - If \(\sin x \in [0, 1)\), then \(\lfloor \sin x \rfloor = 0\).
  - If \(\sin x \in [-1, 0)\), then \(\lfloor \sin x \rfloor = -1\).
  - At \(\sin x = 1\), \(\lfloor \sin x \rfloor = 0\).

### Step 2: Analyze the possible values of \(\lfloor \cos x \rfloor\)
- The cosine function \(\cos x\) also takes values in the interval \([-1, 1]\) for all \(x\).
- Therefore, \(\lfloor \cos x \rfloor\) will also take values from the set \(\{-1, 0\}\), because:
  - If \(\cos x \in [0, 1)\), then \(\lfloor \cos x \rfloor = 0\).
  - If \(\cos x \in [-1, 0)\), then \(\lfloor \cos x \rfloor = -1\).
  - At \(\cos x = 1\), \(\lfloor \cos x \rfloor = 0\).

### Step 3: Possible values of \(f(x)\)
Now we combine the two possible values from \(\lfloor \sin x \rfloor\) and \(\lfloor \cos x \rfloor\). The sum \(f(x) = \lfloor \sin x \rfloor + \lfloor \cos x \rfloor\) can take the following values:

- If \(\lfloor \sin x \rfloor = 0\) and \(\lfloor \cos x \rfloor = 0\), then \(f(x) = 0 + 0 = 0\).
- If \(\lfloor \sin x \rfloor = -1\) and \(\lfloor \cos x \rfloor = 0\), then \(f(x) = -1 + 0 = -1\).
- If \(\lfloor \sin x \rfloor = 0\) and \(\lfloor \cos x \rfloor = -1\), then \(f(x) = 0 + (-1) = -1\).
- If \(\lfloor \sin x \rfloor = -1\) and \(\lfloor \cos x \rfloor = -1\), then \(f(x) = -1 + (-1) = -2\).

Thus, the possible values for \(f(x)\) are \( \{-2, -1, 0\} \).

### Final Answer:  
The range of \(f(x)\) is \(\{-2, -1, 0\}\).

Do you need any more details or have further questions?

Here are some related questions to expand the topic:
1. What is the range of \( f(x) = \lfloor \sin x \rfloor + \lfloor \tan x \rfloor \)?
2. How does the range of \( f(x) = \lfloor \sin x \rfloor \times \lfloor \cos x \rfloor \) change?
3. What is the behavior of \( f(x) = \lceil \sin x \rceil + \lfloor \cos x \rfloor \), where \( \lceil . \rceil \) is the ceiling function?
4. How does the range change if \( f(x) = [x] + \sin(x) \)?
5. What would be the maximum value of \( \lfloor \sin x \rfloor + \lfloor \cos x \rfloor \) on the interval \( [0, 2\pi] \)?

**Tip:** The greatest integer function \(\lfloor x \rfloor\) returns the largest integer less than or equal to \(x\), which simplifies many trigonometric problems!

Explore how to determine the range of the function f(x) = ⌊sin x⌋ + ⌊cos x⌋, where the greatest integer function (⌊x⌋) is applied to sine and cosine functions. Suitable for students in Grades 10-12, this problem covers key concepts in trigonometry and the floor function.

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