Final answer:
To find the inverse function of A(x) = -4/(3x+5), one must replace A(x) with y, switch x and y, and then solve for the new y. After following a series of algebraic steps, we find that the inverse function is A^-1(x) = (-5x - 4) / (3x).
Step-by-step explanation:
To find the inverse of the function A(x) = \frac{-4}{3x+5}, we need to swap the roles of x and A(x), and then solve for the new x. Let's denote the inverse function as A-1(x) and follow these steps:
Replace A(x) with y for convenience: y = \frac{-4}{3x+5}.
Swap x and y: x = \frac{-4}{3y+5}.
Solve for y: Multiply both sides by (3y+5) to get x(3y+5) = -4, which simplifies to 3xy + 5x = -4.
Isolate the term with y: 3xy = -4 - 5x.
Factor out y: y(3x) = -5x - 4.
Divide both sides by 3x to solve for y: y = \frac{-5x - 4}{3x}.
Therefore, the inverse function of A(x) is A-1(x) = \frac{-5x - 4}{3x}.