Solution

As per the typical strategy for finding inverse functions, we will replace the function with \(y\), then solve for the input. The resulting equation will be the inverse function. \[ \solve{ y&=&3\ln(2x-3)\\ \frac{y}{3}&=&\ln(2x-3)\\ e^{\frac{y}{3}}&=&2x-3\\ 3+e^{\frac{y}{3}}&=&2x\\ x&=&\frac{1}{2}\left(3+e^{\frac{y}{3}}\right)\\ f^{-1}(y)&=&\frac{1}{2}\left(3+e^{\frac{y}{3}}\right) } \] Depending on how the question is asked, you may consider swapping the \(x\) and \(y\) variables to rewrite the inverse function like this: \[ f^{-1}(x)=\frac{1}{2}\left(3+e^{\frac{x}{3}}\right) \] You may also simplify the result into various different iterations, but I will leave my answer here.