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