Answer
$f(^{-1}(x)=\dfrac{5x+1}{2x+2},\text{ }x\ne-1$
Work Step by Step
Let $y=f(x)$. Then the given one-to-one function, $
f(x)=\dfrac{-2x+1}{2x-5},\text{ }x\ne\dfrac{5}{2}
,$ becomes
\begin{align*}\require{cancel}
y&=\dfrac{-2x+1}{2x-5},\text{ }x\ne\dfrac{5}{2}
.\end{align*}
To find the inverse, interchange the $x$ and $y$ variables and then solve for $y$. That is,
\begin{align*}
x&=\dfrac{-2y+1}{2y-5}
&(\text{interchange $x$ and $y$})
\\\\
(2y-5)(x)&=\left(\dfrac{-2y+1}{\cancel{2y-5}}\right)(\cancel{2y-5})
&(\text{solve for $y$})
\\\\
2xy-5x&=-2y+1
\\
2xy+2y&=5x+1
\\
2y(x+1)&=5x+1
\\\\
\dfrac{\cancel2y(\cancel{x+1})}{\cancel2(\cancel{x+1})}&=\dfrac{5x+1}{2(x+1)}
\\\\
y&=\dfrac{5x+1}{2x+2}
.\end{align*}
Hence, the inverse of $
f(x)=\dfrac{-2x+1}{2x-5},\text{ }x\ne\dfrac{5}{2}
$ is $
f(^{-1}(x)=\dfrac{5x+1}{2x+2},\text{ }x\ne-1
$.
