Answer
$f^{-1}(x)=-\dfrac{x+2}{3x-4}$
Work Step by Step
$f(x)=\dfrac{4x-2}{3x+1}$
Rewrite this expression as $y=\dfrac{4x-2}{3x+1}$ and solve for $x$:
$y=\dfrac{4x-2}{3x+1}$
Take $3x+1$ to multiply the left side:
$y(3x+1)=4x-2$
$3xy+y=4x-2$
Take $4x$ to the left side and $y$ to the right side:
$3xy-4x=-2-y$
Take common factor $x$ from the left side:
$x(3y-4)=-2-y$
Take $3y-4$ to divide the right side:
$x=\dfrac{-2-y}{3y-4}$
$x=-\dfrac{y+2}{3y-4}$
Interchange $x$ and $y$:
$y=-\dfrac{x+2}{3x-4}$
The inverse of the original function is $f^{-1}(x)=-\dfrac{x+2}{3x-4}$