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