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's how to solve the problem: The function is defined as f(x) = (72)/(px+r). a) The values of p and r. Step 1: Use the given conditions to form a system of equations. Given f(6) = 12: 12 = (72)/(p(6)+r) 12(6p+r) = 72 6p+r = (72)/(12) 6p+r = 6 (Equation 1) Given f(7) = 9: 9 = (72)/(p(7)+r) 9(7p+r) = 72 7p+r = (72)/(9) 7p+r = 8 (Equation 2) Step 2: Solve the system of linear equations. Subtract Equation 1 from Equation 2: (7p+r) - (6p+r) = 8 - 6 p = 2 Substitute p=2 into Equation 1: 6(2)+r = 6 12+r = 6 r = 6-12 r = -6 The values are p = 2 and r = -6. b) f^-1((5)/(3)) Step 1: Find the inverse function f^-1(x). Using p=2 and r=-6, the function is f(x) = (72)/(2x-6). Let y = f(x). So, y = (72)/(2x-6). To find the inverse, swap x and y: x = (72)/(2y-6) Solve for y: x(2y-6) = 72 2y-6 = (72)/(x) 2y = (72)/(x) + 6 y = (36)/(x) + 3 So, f^-1(x) = (36)/(x) + 3. Step 2: Evaluate f^-1((5)/(3)). Substitute x = (5)/(3) into the inverse function: f^-1((5)/(3)) = (36)/(5)3 + 3 f^-1((5)/(3)) = 36 × (3)/(5) + 3 f^-1((5)/(3)) = (108)/(5) + 3 f^-1((5)/(3)) = (108)/(5) + (15)/(5) f^-1((5)/(3)) = (123)/(5) The value is f^-1((5)/(3)) = (123)/(5). That's 2 down. 3 left today — send the next one.