This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back ISRAEL — missed you this week. Here are the domains for the given functions: a) Find the domain of z = (x^2 + y^2) Step 1: Identify the restriction for the natural logarithm function. The natural logarithm (u) is defined only when its argument u is strictly positive. Therefore, we must have x^2 + y^2 > 0. Step 2: Determine the values of (x, y) that satisfy the inequality. The expression x^2 + y^2 is always non-negative. It is equal to zero only when x=0 and y=0 simultaneously. For all other points (x, y), x^2 + y^2 will be positive. So, the condition x^2 + y^2 > 0 means that (x, y) can be any point in R^2 except for the origin (0, 0). The domain is \(x, y) R^2 (x, y) (0, 0)\. b) Find the domain of f(x, y) = sqrt(x^2 + y^2 - 2)y - 2 Step 1: Identify restrictions for the square root. The expression under the square root must be non-negative. x^2 + y^2 - 2 0 x^2 + y^2 2 This condition means that the points (x, y) must lie on or outside the circle centered at the origin with radius sqrt(2). Step 2: Identify restrictions for the denominator. The denominator cannot be zero. y - 2 0 y 2 This condition means that the points (x, y) cannot lie on the line y=2. Step 3: Combine all restrictions to define the domain. The domain consists of all points (x, y) such that x^2 + y^2 2 and y 2. The domain is \(x, y) R^2 x^2 + y^2 2 and y 2\. Send me the next one 📸