Final answer:
To find the inverse of the function f(x) = (4x-2)/(3x+1), switch the roles of x and y and solve for y. The inverse of the function is f^(-1)(x) = -(x+2)/(3x-4).
Step-by-step explanation:
To find the inverse of the function f(x) = (4x-2)/(3x+1), we need to switch the roles of x and y and solve for y.
Step 1: Replace f(x) with y. The equation becomes x = (4y-2)/(3y+1).
Step 2: Multiply both sides of the equation by (3y+1) to eliminate the denominator. This gives us x(3y+1) = 4y-2.
Step 3: Expand the equation and arrange it in the form of a quadratic equation: 3xy + x = 4y - 2. Rearrange the terms: 3xy - 4y = -x - 2.
Step 4: Factor out y on the left side of the equation and factor out -1 on the right side: y(3x-4) = -(x+2).
Step 5: Divide both sides of the equation by (3x-4) to isolate y: y = -(x+2)/(3x-4).
Therefore, the inverse of the function f(x) = (4x-2)/(3x+1) is f^(-1)(x) = -(x+2)/(3x-4).