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 RUDDY — missed you this week. a) To express the function g(f(x)), we substitute f(x) into g(x). Step 1: Substitute f(x) into g(x). Given f(x) = (2x+1)/(3x+3) and g(x) = 3x+2. g(f(x)) = g((2x+1)/(3x+3)) = 3((2x+1)/(3x+3)) + 2 Step 2: Simplify the expression. g(f(x)) = (3(2x+1))/(3(x+1)) + 2 g(f(x)) = (2x+1)/(x+1) + 2 To combine the terms, find a common denominator: g(f(x)) = (2x+1)/(x+1) + (2(x+1))/(x+1) g(f(x)) = (2x+1 + 2x+2)/(x+1) g(f(x)) = (4x+3)/(x+1) Step 3: Determine the domain. The domain of g(f(x)) requires x to be in the domain of f, and f(x) to be in the domain of g. The domain of f(x) is x R, x ≠ -1. The domain of g(x) is x R. Since f(x) will always produce a real number (for x ≠ -1), and g(x) accepts all real numbers, the only restriction is x ≠ -1. Therefore, the domain of g(f(x)) is x R, x ≠ -1. The function g(f(x)) expressed in a similar manner is: g(f(x)): x (4x+3)/(x+1), x R, x ≠ -1 b) An element x is invariant under a function g if g(x) = x. Step 1: Set up the equation for an invariant element. Given g(x) = 3x+2. We need to solve g(x) = x: 3x+2 = x Step 2: Solve for x. Subtract x from both sides: 3x - x + 2 = 0 2x + 2 = 0 Subtract 2 from both sides: 2x = -2 Divide by 2: x = -1 Step 3: Verify the element is in the domain of g. The domain of g is x R. Since -1 is a real number, it is in the domain of g. The element in the domain of g which is invariant under g is: x = -1 Send me the next one 📸