If f(x) = n , where n is an integer such that `n le x lt n +1`, the range of f(x) is

Options

A. the set of all real numbers
B. the set of all positive integers
C. the set of all integers
D. the set of all negative integers

Explanation

To solve the problem, we need to analyze the function \( f(x) = n \), where \( n \) is an integer such that \( n \leq x < n + 1 \). We want to determine the range of \( f(x) \).

### Step-by-Step Solution:

1. **Understanding the Function**:
The function \( f(x) = n \) means that for any value of \( x \) that falls within the interval \( [n, n + 1) \), the output of the function is the integer \( n \).

2. **Identifying the Intervals**:
For each integer \( n \):
- If \( n = 0 \), then \( 0 \leq x < 1 \) implies \( f(x) = 0 \).
- If \( n = 1 \), then \( 1 \leq x < 2 \) implies \( f(x) = 1 \).
- If \( n = -1 \), then \( -1 \leq x < 0 \) implies \( f(x) = -1 \).
- This pattern continues for all integers.

3. **Determining Possible Values of \( n \)**:
Since \( n \) can be any integer, we can conclude that for every integer \( n \), there exists a corresponding interval \( [n, n + 1) \) where \( f(x) = n \).

4. **Conclusion on the Range**:
Since \( n \) can take any integer value, the range of \( f(x) \) is the set of all integers. Thus, the range of \( f(x) \) is \( \mathbb{Z} \) (the set of all integers).

### Final Answer:
The range of \( f(x) \) is the set of all integers, denoted as \( \mathbb{Z} \).

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