Define greatest integer function. Write its domain and range.

Answer

### Step-by-Step Solution:

1. **Definition of the Greatest Integer Function**:
The greatest integer function, often denoted as \( \lfloor x \rfloor \), is defined as the largest integer that is less than or equal to \( x \). In other words, for any real number \( x \), \( \lfloor x \rfloor \) gives you the integer part of \( x \) by rounding it down to the nearest integer.

2. **Domain of the Greatest Integer Function**:
The domain of the greatest integer function is all real numbers. This means that you can input any real number \( x \) into the function. Therefore, the domain can be expressed in interval notation as:
\[ \text{Domain} = (-\infty, +\infty) \]

3. **Range of the Greatest Integer Function**:
The range of the greatest integer function consists of all integers. Since the function outputs the greatest integer less than or equal to \( x \), for any real number input, the output will always be an integer. Thus, the range can be expressed as:
\[ \text{Range} = \{ n \in \mathbb{Z} \} \quad \text{(where } \mathbb{Z} \text{ is the set of all integers)} \]

### Summary:
- **Greatest Integer Function**: \( \lfloor x \rfloor \) is the largest integer less than or equal to \( x \).
- **Domain**: \( (-\infty, +\infty) \)
- **Range**: All integers \( \mathbb{Z} \)