For the function F (x) = `[(1)/([x])]` , where [x] denotes the greatest integer less than or equal to x, which of the following statements are true?

Options

A. the domain is `(-infty,infty)`
B. the range is {0} `cup` {-1} `cup` {1}
C. the domain is `(-infty,0) cup [1,infty)`
D. the range is {0} `cup` {1}

Answer

To solve the problem regarding the function \( F(x) = \frac{1}{[\![x]\!]} \), where \([\![x]\!]\) denotes the greatest integer less than or equal to \( x \), we will analyze the domain and range of the function step by step.

### Step 1: Understanding the Greatest Integer Function
The greatest integer function \([\![x]\!]\) returns the largest integer less than or equal to \( x \). For example:
- \([\![2.3]\!] = 2\)
- \([\![3]\!] = 3\)
- \([\![-1.5]\!] = -2\)

### Step 2: Determine When the Function is Undefined
The function \( F(x) \) will be undefined when the denominator \([\![x]\!]\) is equal to 0. This occurs when:
\[ [\![x]\!] = 0 \]
The greatest integer function \([\![x]\!]\) is 0 when \( x \) is in the interval \( [0, 1) \). Therefore, the function \( F(x) \) is undefined for:
\[ x \in [0, 1) \]

### Step 3: Establishing the Domain
The domain of \( F(x) \) includes all real numbers except those in the interval \( [0, 1) \). Thus, we can express the domain as:
\[ \text{Domain} = (-\infty, 0) \cup [1, \infty) \]

### Step 4: Analyzing the Range
Next, we will analyze the range of the function \( F(x) \):
1. For \( x < 0 \):
- Here, \([\![x]\!]\) will be a negative integer. Thus, \( F(x) = \frac{1}{[\![x]\!]} \) will yield negative values.
- For example, if \( x = -0.5 \), then \([\![x]\!] = -1\) and \( F(-0.5) = -1\).

2. For \( x \geq 1 \):
- Here, \([\![x]\!]\) will be a positive integer. Thus, \( F(x) = \frac{1}{[\![x]\!]} \) will yield positive values.
- For example, if \( x = 1 \), then \([\![x]\!] = 1\) and \( F(1) = 1\).
- If \( x = 2 \), then \([\![x]\!] = 2\) and \( F(2) = \frac{1}{2}\).

### Step 5: Identifying Possible Values of \( F(x) \)
From the analysis:
- As \( x \) approaches 0 from the left, \( F(x) \) approaches negative infinity.
- As \( x \) approaches 1 from the right, \( F(x) \) approaches 1.
- For \( x \) greater than 1, \( F(x) \) will yield values between 0 and 1 (exclusive) as well as 1 itself.

Thus, the range of \( F(x) \) includes:
\[ \text{Range} = \{0, 1\} \cup (-\infty, 0) \]

### Conclusion
Based on our analysis:
- The domain of \( F(x) \) is \( (-\infty, 0) \cup [1, \infty) \).
- The range of \( F(x) \) is \( (-\infty, 0) \cup \{1\} \).

### Final Answer
The correct statements regarding the function \( F(x) \) are:
- Domain: \( (-\infty, 0) \cup [1, \infty) \)
- Range: \( (-\infty, 0) \cup \{1\} \)