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 determine the domain of the function \( f(x) = \sqrt{4 - \lfloor x \rfloor} \), we need to ensure that the expression under the square root is non-negative. Thus, we must have: \[ 4 - \lfloor x \rfloor \geq 0 \] This simplifies to: \[ \lfloor x \rfloor \leq 4 \] The greatest integer function \( \lfloor x \rfloor \) takes integer values, so the possible values of \( \lfloor x \rfloor \) can be \( \ldots, -1, 0, 1, 2, 3, 4 \). Hence, \( x \) must be at least \( -2 \) (where \( \lfloor -2 \rfloor = -2 \)), and it can be at most \( 3.999 \) (where \( \lfloor 4 \rfloor = 4 \)). This gives us the domain \( -2 \leq x < 4 \). Now, analyzing the answer choices: - Option (a) is incorrect because it includes the endpoint of 3, but we actually see that \( \lfloor 3 \rfloor = 3 \) keeps it valid. - Option (b) is correct and includes both endpoints. - Option (c) is incorrect because it improperly excludes -2. - Option (d) is also correct since it excludes only option c. Thus, the correct answer is that the domain is \( -2 \leq x < 4 \), making the answer (b) -2 ≤ x < 3 is the option that fits best. So, the selected answer is (b): -2 ≤ x < 3! Here’s an extra sprinkle of fun knowledge: Did you know that the floor function, \( \lfloor x \rfloor \), can be handy when dealing with computer programming? It helps in rounding down to the nearest integer, which can be useful in scenarios like managing array indexes! Additionally, if you're intrigued by more nuanced mathematical functions like the floor function, exploring topics like the ceiling function \( \lceil x \rceil \) or modulus might tantalize your curiosity. Not only do they have unique applications, but they also interconnect beautifully in various mathematical proofs!