Answer
$f^{-1}(x)=\frac{3x+1}{x-2}$
Work Step by Step
Step 1. Replace $f(x)$ with $y$ to get $y=\frac{2x+1}{x-3}$
Step 2. Exchange $x,y$ to get $x=\frac{2y+1}{y-3}$
Step 3. Solving for $y$, we have $xy-3x=2y+1$, $(x-2)y=3x+1$; thus $y=\frac{3x+1}{x-2}$
Step 4. Replace $y$ with $f^{-1}(x)$; we have $f^{-1}(x)=\frac{3x+1}{x-2}$
