Check for identical`f(x)=[{x}],g(x)={[x]} [Note that f(x) and g(x) are constant functions]

Answer

To determine whether the functions \( f(x) = \lfloor x \rfloor \) (greatest integer function) and \( g(x) = \{x\} \) (fractional part function) are identical, we will follow a systematic approach.

### Step-by-Step Solution

1. **Understanding the Functions**:
- The function \( f(x) = \lfloor x \rfloor \) gives the greatest integer less than or equal to \( x \). This function is defined for all real numbers and always returns an integer.
- The function \( g(x) = \{x\} = x - \lfloor x \rfloor \) gives the fractional part of \( x \). This function is defined for all real numbers and returns a value in the range \( [0, 1) \).

2. **Identical Functions Definition**:
- Two functions \( f \) and \( g \) are said to be identical if:
1. The domain of \( f \) is equal to the domain of \( g \).
2. The range of \( f \) is equal to the range of \( g \).
3. For all \( x \) in the domain, \( f(x) = g(x) \).

3. **Finding the Domain**:
- The domain of both functions \( f(x) \) and \( g(x) \) is all real numbers \( \mathbb{R} \).

4. **Finding the Range**:
- For \( f(x) = \lfloor x \rfloor \), the range is all integers \( \mathbb{Z} \).
- For \( g(x) = \{x\} \), the range is \( [0, 1) \).

5. **Comparing the Ranges**:
- The range of \( f(x) \) is \( \mathbb{Z} \) (all integers), while the range of \( g(x) \) is \( [0, 1) \). Since these two ranges are not equal, we can conclude that \( f(x) \) and \( g(x) \) are not identical.

6. **Conclusion**:
- Since the ranges of \( f(x) \) and \( g(x) \) are different, we can conclude that \( f(x) \) and \( g(x) \) are not identical functions.

### Final Answer:
The functions \( f(x) = \lfloor x \rfloor \) and \( g(x) = \{x\} \) are **not identical**.