Answer
${{g}^{-1}}\left( x \right)=\frac{2x-1}{3}$.
Work Step by Step
$g\left( x \right)=\frac{3x+1}{2}$
Evaluate the inverse of the function $g\left( x \right)=\frac{3x+1}{2}$ as follows.
Replace the function $f\left( x \right)$ with $y$.
$y=\frac{3x+1}{2}$
Interchange the variables x and y.
$x=\frac{3y+1}{2}$
Solve for the value of y.
$\begin{align}
& x=\frac{3y+1}{2} \\
& 2x=3y+1 \\
& 3y=2x-1 \\
& y=\frac{2x-1}{3}
\end{align}$
Replace y with ${{g}^{-1}}\left( x \right)$ as follows.
${{g}^{-1}}\left( x \right)=\frac{2x-1}{3}$
