Final answer:
To find the inverse of f(x) = (3x - 4) / (2x + 1), we switch x and y, solve for y, and find f^-1(x) = (-x - 4) / (2x - 3).
Step-by-step explanation:
To find the inverse f^-1(x) of the function f(x) = (3x - 4) / (2x + 1), we must switch the roles of x and y and then solve for y, which will give us the inverse function. Start by replacing f(x) with y:
y = (3x - 4) / (2x + 1)
Now interchange x and y:
x = (3y - 4) / (2y + 1)
Multiply both sides of the equation by (2y + 1) to get rid of the denominator:
x(2y + 1) = 3y - 4
Expand the left side:
2xy + x = 3y - 4
Rearrange the terms to get all terms involving y on one side and the constant term on the other:
2xy - 3y = -x - 4
Factor out y:
y(2x - 3) = -x - 4
Finally, solve for y:
y = (-x - 4) / (2x - 3)
So, the inverse function is f^-1(x) = (-x - 4) / (2x - 3).