To find the inverse of the given function, isolate x from the equation:
[tex]\begin{gathered} y=\ln (\frac{x+2}{3})+1 \\ \Rightarrow y-1=\ln (\frac{x+2}{3}) \\ \Rightarrow e^{y-1}=\frac{x+2}{3} \\ \Rightarrow3e^{y-1}=x+2 \\ \Rightarrow3e^{y-1}-2=x \\ \\ \therefore x=3e^{y-1}-2 \end{gathered}[/tex]Swap x for y and y for x in the expression for x to find the inverse function:
[tex]y=3e^{x-1}-2[/tex]Therefore, the inverse function of y=ln((x+2)/3)+1 is y=3e^(x-1)-2.
