Find the domain and range of the function `f(x)=[sinx]`

Answer

To find the domain and range of the function \( f(x) = [\sin x] \), where \([\cdot]\) denotes the greatest integer function (also known as the floor function), we can follow these steps:

### Step 1: Determine the Domain
The function \( \sin x \) is defined for all real numbers \( x \). Therefore, the domain of \( \sin x \) is:
\[ \text{Domain of } \sin x = \{ x \in \mathbb{R} \} \]
Since the greatest integer function is also defined for all real numbers, the domain of \( f(x) = [\sin x] \) remains:
\[ \text{Domain of } f(x) = \{ x \in \mathbb{R} \} \]

### Step 2: Determine the Range
Next, we need to find the range of \( f(x) = [\sin x] \). The sine function oscillates between -1 and 1, inclusive. Thus, the range of \( \sin x \) is:
\[ \text{Range of } \sin x = [-1, 1] \]
Now, we apply the greatest integer function to this range. The greatest integer function takes any real number and gives the largest integer less than or equal to that number.

- For \( \sin x = 1 \), the greatest integer is \( 1 \).
- For \( \sin x \) values just below \( 1 \) (e.g., \( 0.999 \)), the greatest integer is still \( 0 \).
- For \( \sin x = 0 \), the greatest integer is \( 0 \).
- For \( \sin x = -1 \), the greatest integer is \( -1 \).
- For \( \sin x \) values just above \( -1 \) (e.g., \( -0.999 \)), the greatest integer is still \( -1 \).

Thus, the only integer values that \( \sin x \) can take when applying the greatest integer function are:
\[ -1, 0, \text{ and } 1 \]
Therefore, the range of \( f(x) = [\sin x] \) is:
\[ \text{Range of } f(x) = \{-1, 0, 1\} \]

### Final Answer
- **Domain**: \( \{ x \in \mathbb{R} \} \)
- **Range**: \( \{-1, 0, 1\} \)