3. what is the domain of \( \mathrm{f}(\mathrm{x})=\sqrt{4-\lfloor x\rfloor} \) ? \( \begin{array}{llll}\text { a. }[-2,3] & \text { b. }-2 \leq x \leq 3 & \text { c. }-2

Ask by Willis Bartlett. in Ethiopia

Jan 03,2025

To find the domain of the function \( f(x) = \sqrt{4 - \lfloor x \rfloor} \), we need to ensure that the expression inside the square root is non-negative, since the square root function is only defined for non-negative values. 1. **Identify the condition for the square root:** \[ 4 - \lfloor x \rfloor \geq 0 \] This can be rearranged to: \[ \lfloor x \rfloor \leq 4 \] 2. **Understanding the floor function:** The floor function \( \lfloor x \rfloor \) gives the greatest integer less than or equal to \( x \). Therefore, \( \lfloor x \rfloor \) can take integer values. The condition \( \lfloor x \rfloor \leq 4 \) implies that \( \lfloor x \rfloor \) can be any integer less than or equal to 4. 3. **Determine the possible integer values for \( \lfloor x \rfloor \):** The integers that satisfy \( \lfloor x \rfloor \leq 4 \) are: \[ \lfloor x \rfloor = \ldots, -2, -1, 0, 1, 2, 3, 4 \] 4. **Translate these integer values back to \( x \):** - If \( \lfloor x \rfloor = 4 \), then \( 4 \leq x < 5 \). - If \( \lfloor x \rfloor = 3 \), then \( 3 \leq x < 4 \). - If \( \lfloor x \rfloor = 2 \), then \( 2 \leq x < 3 \). - If \( \lfloor x \rfloor = 1 \), then \( 1 \leq x < 2 \). - If \( \lfloor x \rfloor = 0 \), then \( 0 \leq x < 1 \). - If \( \lfloor x \rfloor = -1 \), then \( -1 \leq x < 0 \). - If \( \lfloor x \rfloor = -2 \), then \( -2 \leq x < -1 \). 5. **Combine the intervals:** The combined intervals for \( x \) based on the values of \( \lfloor x \rfloor \) are: \[ [-2, -1) \cup [-1, 0) \cup [0, 1) \cup [1, 2) \cup [2, 3) \cup [3, 4) \cup [4, 5) \] 6. **Determine the overall domain:** The overall domain of \( f(x) \) is: \[ [-2, 5) \] Now, let's analyze the options provided: - a. \([-2,3]\) - b. \(-2 \leq x \leq 3\) - c. \(-2 < x < 3\) - d. all except c The correct answer must include all values from \([-2, 3]\) and also include values up to 5, which means options a and c are not correct. The best option that includes the domain is option b, which is \(-2 \leq x \leq 3\). Thus, the correct answer is: **b. \(-2 \leq x \leq 3\)**.