The condition for an inverse function is:
[tex]g(x)=y\longrightarrow g^{-1}(y)=x[/tex]Then, we can calculate:
[tex]\begin{gathered} g(g^{-1}(x))=x \\ \frac{-2g^{-1}}{3}-5=x \\ -\frac{2}{3}\cdot g^{-1}=x+5 \\ g^{-1}=-\frac{3}{2}(x+5) \\ g^{-1}=-\frac{3}{2}x-\frac{15}{2} \end{gathered}[/tex]The inverse of g(x) is:
[tex]g^{-1}(y)=-\frac{3}{2}y-\frac{15}{2}[/tex]