[tex]f(x)=3+2\ln x,\ x > 0\\\\y=3+2\ln x\\\\\text{replace x with y and y with x}\\\\3+2\ln y=x\ \ \ \ |-3\\\\2\ln y=x-3\ \ \ \ |:2\\\\\ln y=\dfrac{y-3}{2}\to e^{\ln y}=e^{\frac{y-3}{2}}\\\\y=e^{\frac{y-3}{2}}\\\\Answer:\ f^{-1}(x)=e^{\frac{y-3}{2}}[/tex]
