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: Replace f(x) with y. The given function is f(x) = (2x-1)/(x+2). Let y = f(x), so we have: y = (2x-1)/(x+2) Step 2: Swap x and y to find the inverse relation. x = (2y-1)/(y+2) Step 3: Solve the equation for y. Multiply both sides by (y+2): x(y+2) = 2y-1 Distribute x on the left side: xy + 2x = 2y - 1 Gather all terms containing y on one side of the equation and all other terms on the other side. Subtract 2y from both sides and subtract 2x from both sides: xy - 2y = -1 - 2x Factor out y from the terms on the left side: y(x - 2) = -1 - 2x Divide by (x-2) to isolate y: y = (-1 - 2x)/(x - 2) This can also be written by multiplying the numerator and denominator by -1: y = (-(1 + 2x))/(-(2 - x)) = (2x+1)/(2-x) Step 4: Replace y with f^-1(x). f^-1(x) = (2x+1)/(2-x) The inverse function is defined for all x such that 2-x ≠ 0, which means x ≠ 2. The final answer is f^-1(x) = (2x+1)/(2-x). Got more? Send 'em!