Answer
$f(^{-1}(x)=\dfrac{-4x-10}{x-5},\text{ }x\ne5$
Work Step by Step
Let $y=f(x)$. Then the given one-to-one function, $
f(x)=\dfrac{5x-10}{x+4},\text{ }x\ne-4
,$ becomes
\begin{align*}\require{cancel}
y&=\dfrac{5x-10}{x+4},\text{ }x\ne-4
.\end{align*}
To find the inverse, interchange the $x$ and $y$ variables and then solve for $y$. That is,
\begin{align*}
x&=\dfrac{5y-10}{y+4}
&(\text{interchange $x$ and $y$})
\\\\
(y+4)(x)&=\left(\dfrac{5y-10}{\cancel{y+4}}\right)(\cancel{y+4})
&(\text{solve for $y$})
\\\\
xy+4x&=5y-10
\\
xy-5y&=-4x-10
\\\\
y(x-5)&=-4x-10
\\\\
\dfrac{y(\cancel{x-5})}{\cancel{x-5}}&=\dfrac{-4x-10}{x-5}
\\\\
y&=\dfrac{-4x-10}{x-5}
.\end{align*}
Hence, the inverse of $
f(x)=\dfrac{5x-10}{x+4},\text{ }x\ne-4
$ is $
f(^{-1}(x)=\dfrac{-4x-10}{x-5},\text{ }x\ne5
$.
