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
Step 1: Find f^-1(x) (Part a). Let y = f(x), so y = (3x - 2)/(5x + 4). To find the inverse function, swap x and y: x = (3y - 2)/(5y + 4) Multiply both sides by (5y + 4): x(5y + 4) = 3y - 2 5xy + 4x = 3y - 2 Rearrange the terms to isolate y: 5xy - 3y = -2 - 4x Factor out y: y(5x - 3) = -2 - 4x Solve for y: y = (-2 - 4x)/(5x - 3) This can be written as: f^-1(x) = (4x + 2)/(3 - 5x) Step 2: Find g^-1(x) (Part b). Let y = g(x), so y = (7x + 2)/(4x - 3). To find the inverse function, swap x and y: x = (7y + 2)/(4y - 3) Multiply both sides by (4y - 3): x(4y - 3) = 7y + 2 4xy - 3x = 7y + 2 Rearrange the terms to isolate y: 4xy - 7y = 2 + 3x Factor out y: y(4x - 7) = 2 + 3x Solve for y: y = (2 + 3x)/(4x - 7) So, g^-1(x) = (3x + 2)/(4x - 7) Step 3: Find the values of x if g^-1(x) = 1 - f^-1(x) (Part c). Substitute the expressions for f^-1(x) and g^-1(x) into the equation: (3x + 2)/(4x - 7) = 1 - (4x + 2)/(3 - 5x) First, simplify the right side: 1 - (4x + 2)/(3 - 5x) = (3 - 5x)/(3 - 5x) - (4x + 2)/(3 - 5x) = ((3 - 5x) - (4x + 2))/(3 - 5x) = (3 - 5x - 4x - 2)/(3 - 5x) = (1 - 9x)/(3 - 5x) Now the equation is: (3x + 2)/(4x - 7) = (1 - 9x)/(3 - 5x) Cross-multiply: (3x + 2)(3 - 5x) = (1 - 9x)(4x - 7) Expand both sides: 9x - 15x^2 + 6 - 10x = 4x - 7 - 36x^2 + 63x -15x^2 - x + 6 = -36x^2 + 67x - 7 Move all terms to one side to form a quadratic equation: -15x^2 + 36x^2 - x - 67x + 6 + 7 = 0 21x^2 - 68x + 13 = 0 Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a with a=21, b=-68, c=13: x = -(-68) ± sqrt((-68)^2 - 4(21)(13))2(21) x = 68 ± sqrt(4624 - 1092)42 x = 68 ± sqrt(3532)42 Calculate the square root: sqrt(3532) ≈ 59.43063 Now find the two values for x: x_1 = (68 + 59.43063)/(42) = (127.43063)/(42) ≈ 3.03406 x_2 = (68 - 59.43063)/(42) = (8.56937)/(42) ≈ 0.20398 Rounding to two decimal places: x_1 ≈ 3.03 x_2 ≈ 0.20 The values of x are 3.03 and 0.20. That's 2 down. 3 left today — send the next one.