Choose the correct domain. A. \( (-5,3) \) B. \( (-10,10) \) C. \( (-4,0) \) D. \( (-\infty, \infty) \) What is the set of all x-values such that \( f(x)=-1 \) ? A. \( (-5,3) \) B. \( \varnothing \) C. \( \{1\} \)

It appears you've provided two separate questions, each with its own set of multiple-choice options. To assist you accurately, I'll address each question individually. However, please note that for complete and precise answers, additional information about the function \( f(x) \) is necessary, especially for determining the set of all \( x \)-values such that \( f(x) = -1 \). Let's break down each question: --- ### **1. Choosing the Correct Domain** **Options:** - **A.** \( (-5, 3) \) - **B.** \( (-10, 10) \) - **C.** \( (-4, 0) \) - **D.** \( (-\infty, \infty) \) **Understanding the Domain:** The domain of a function \( f(x) \) is the set of all possible \( x \)-values for which the function is defined. Without the specific function \( f(x) \), it's challenging to determine the exact domain. However, if you provide the function, I can help identify its domain. **Example:** - **If \( f(x) = \sqrt{x} \):** The domain is \( [0, \infty) \) because the square root of a negative number is not a real number. - **If \( f(x) = \frac{1}{x} \):** The domain is \( (-\infty, 0) \cup (0, \infty) \) because division by zero is undefined. **Please provide the function \( f(x) \) to determine the correct domain from the given options.** --- ### **2. Finding \( x \)-Values Such That \( f(x) = -1 \)** **Options:** - **A.** \( (-5, 3) \) - **B.** \( \varnothing \) (Empty Set) - **C.** \( \{1\} \) **Understanding the Solution Set:** To find all \( x \)-values where \( f(x) = -1 \), you need to solve the equation \( f(x) = -1 \). The solution set depends entirely on the form of \( f(x) \). **Possible Scenarios:** - **If \( f(x) = x - 1 \):** \[ x - 1 = -1 \implies x = 0 \] So, the solution set is \( \{0\} \), which isn't among the provided options. - **If \( f(x) = (x - 1)^2 \):** \[ (x - 1)^2 = -1 \] Since a square is always non-negative, there are no real solutions. Thus, the solution set is \( \varnothing \). - **If \( f(x) = \sin(x) \):** \[ \sin(x) = -1 \implies x = \frac{3\pi}{2} + 2k\pi, \quad k \in \mathbb{Z} \] The solution set in this case is infinite and doesn't match any single-element or interval option provided. **Conclusion:** Without the specific form of \( f(x) \), it's not possible to definitively choose the correct option. If \( f(x) \) is such that \( f(x) = -1 \) has no real solutions, then **Option B (\( \varnothing \))** would be correct. If there's exactly one solution, and it happens to be \( x = 1 \), then **Option C (\( \{1\} \))** would be appropriate. --- **Please provide the function \( f(x) \)** or additional context so that I can offer a more accurate and detailed answer to both questions.