Question

If \( f(x) = \lfloor x \rfloor \), where \( \lfloor x \rfloor \) denotes the greatest integer function, and if the domain of \( f \) is \( \{-3.01, 2.99\} \), then the range of \( f \) is

• A. \( \{-3, 3\} \)
• B. \( \{-4, 3\} \)
• C. \( \{-3, 2\} \)
• D. \( \{-4, 2\} \)
• E. \( \{-2, 3\} \)

Hint:

The greatest integer function, \( \lfloor x \rfloor \), returns the greatest integer less than or equal to \( x \).

Answer 1

The Correct Option is D

Step 1: Understand the Greatest Integer Function The greatest integer function \( \lfloor x \rfloor \) returns the largest integer less than or equal to \( x \). 
For example: \[ \lfloor 3.7 \rfloor = 3, \quad \lfloor -2.3 \rfloor = -3. \] 
Step 2: Evaluate \( f(x) \) for Each Element in the Domain The domain of \( f \) is \( \{-3.01, 2.99\} \). We evaluate \( f(x) = \lfloor x \rfloor \) for each element in the domain: 
1. For \( x = -3.01 \): \[ f(-3.01) = \lfloor -3.01 \rfloor = -4. \] (Since \(-4\) is the greatest integer less than or equal to \(-3.01\).) 
2. For \( x = 2.99 \): \[ f(2.99) = \lfloor 2.99 \rfloor = 2. \] (Since \(2\) is the greatest integer less than or equal to \(2.99\).) 
Step 3: Determine the Range The range of \( f \) is the set of all output values of \( f(x) \). From Step 2, the outputs are \(-4\) and \(2\). 
Therefore, the range is: \[ \{-4, 2\}. \] 
Step 4: Verify the Answer The range \( \{-4, 2\} \) corresponds to option (D). 
Final Answer: The range of \( f \) is: \[ \boxed{\{-4, 2\}}. \] 
Thus, the correct option is (D).