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
Here are the values of x for the given conditions. Given functions: f(x) = 2x^2 - (3x)/(4) g(x) = (x+1)/(1-2x) i) Find the values of x for which g(x-1) = f(1). Step 1: Calculate f(1). Substitute x=1 into f(x): f(1) = 2(1)^2 - (3(1))/(4) = 2 - (3)/(4) = (8)/(4) - (3)/(4) = (5)/(4) Step 2: Calculate g(x-1). Substitute x-1 into g(x): g(x-1) = ((x-1)+1)/(1-2(x-1)) = (x)/(1-2x+2) = (x)/(3-2x) Step 3: Set g(x-1) = f(1) and solve for x. (x)/(3-2x) = (5)/(4) Cross-multiply: 4x = 5(3-2x) 4x = 15 - 10x Add 10x to both sides: 4x + 10x = 15 14x = 15 Divide by 14: x = (15)/(14) The value of x for g(x-1) = f(1) is (15)/(14). ii) Find the values of x for which f(x+1) = 0. Step 1: Calculate f(x+1). Substitute x+1 into f(x): f(x+1) = 2(x+1)^2 - (3(x+1))/(4) Step 2: Set f(x+1) = 0 and solve for x. 2(x+1)^2 - (3(x+1))/(4) = 0 Multiply the entire equation by 4 to eliminate the fraction: 8(x+1)^2 - 3(x+1) = 0 Factor out the common term (x+1): (x+1)[8(x+1) - 3] = 0 Simplify the expression inside the square brackets: (x+1)[8x + 8 - 3] = 0 (x+1)[8x + 5] = 0 This equation holds true if either factor is zero: x+1 = 0 or 8x+5 = 0 Solving the first equation: x = -1 Solving the second equation: 8x = -5 x = -(5)/(8) The values of x for which f(x+1) = 0 are -1, -(5)/(8). Send me the next one 📸