Answer
$f^{-1}(x)=\frac{5x+3}{1-2x}$
Work Step by Step
We are given:
$f(x)=\displaystyle \frac{x-3}{2x+5}$
To find the inverse of $f$, we switch $x$ and $y$ and solve for $y$:
$y= \frac{x-3}{2x+5}$
$x= \frac{y-3}{2y+5}$
$x(2y+5)=y-3$
$2xy+5x=y-3$
$5x+3=y-2xy$
$5x+3=y(1-2x)$
$y=\frac{5x+3}{1-2x}$
Therefore, the inverse is:
$f^{-1}(x)=\frac{5x+3}{1-2x}$
