If the functions $f\left(x\right)$ and $g\left(x\right)$ are defined as $f\left(x\right)=3x-4$,$g\left(x\right)=2+3x$ for all $x\in R$ respectively then $g^{-1}\left(f^{-1}\left(5\right)\right)$ is equal to

The given functions are $f\left(x\right)=3x-4$ and $g\left(x\right)=2+3x$

First find the inverse functions to the above functions

$\begin{array}{rcl}y& =& 3x-4\\ 3x& =& y+4\\ x& =& \frac{y+4}{3}\end{array}$

Hence, $f^{-1}\left(x\right)=\frac{x+4}{3}$

$\begin{array}{rcl}y& =& 2+3x\\ 3x& =& y-2\\ x& =& \frac{y-2}{3}\end{array}$

Hence, $g^{-1}\left(x\right)=\frac{x-2}{3}$

Consider $g^{-1}\left(f^{-1}\left(5\right)\right)$

$\begin{array}{rcl}{g}^{-1}\left(f^{-1}\left(5\right)\right)& =& g^{-1}\left(\frac{5+4}{3}\right)\\ & =& g^{-1}\left(\frac{9}{3}\right)\\ & =& g^{-1}\left(3\right)\\ & =& \frac{3-2}{3}\\ & =& \frac{1}{3}\end{array}$