Answer
The composites ${{\left( f\circ g \right)}^{-1}}\left( x \right)$ and $\left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( x \right)$ with the functions $f\left( x \right)=3x,g\left( x \right)=x+5$ are $\left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( x \right)=\frac{x}{3}-5$ and ${{\left( f\circ g \right)}^{-1}}\left( x \right)=\frac{x}{3}-5$.
Work Step by Step
Assume the functions below:
$\begin{align}
& \left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right) \\
& \left( f\circ g \right)\left( x \right)=3\left( x+5 \right) \\
& \left( f\circ g \right)\left( x \right)=3x+15
\end{align}$
To find ${{f}^{-1}}\left( x \right)$:
Consider the function $f\left( x \right)=3x$
Put $f\left( x \right)=y$,
$y=3x$
Now, put $x\text{ and } y$,
$x=3y$
Solve for the value of $y$,
$y=\frac{x}{3}$
Now, put $y={{f}^{-1}}\left( x \right)$
Here, ${{f}^{-1}}\left( x \right)=\frac{x}{3}$
To find ${{g}^{-1}}\left( x \right)$ :
Let the function below be:
$g\left( x \right)=x+5$
So,
y = x + 5
Now, put x = y + 5
$y=x-5$
Here,
${{g}^{-1}}\left( x \right)=x-5$
Now, to find ${{\left( f\circ g \right)}^{-1}}\left( x \right)$:
The function ${{\left( f\circ g \right)}^{-1}}\left( x \right)$ is the inverse of $\left( f\circ g \right)\left( x \right)$
Therefore,
$y=3x+15$
\[x=3y+15\]
$y=\frac{x-15}{3}$
Now, the following function:
$\begin{align}
& {{\left( f\circ g \right)}^{-1}}\left( x \right)=\frac{x-15}{3} \\
& {{\left( f\circ g \right)}^{-1}}\left( x \right)=\frac{x}{3}-5 \\
\end{align}$
Now, to solve for $\left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( x \right)$
$\begin{align}
& \left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( x \right)={{g}^{-1}}\left( {{f}^{-1}}\left( x \right) \right) \\
& \left( {{g}^{-1}}\circ {{f}^{-1}} \right)\left( x \right)=\frac{x}{3}-5 \\
\end{align}$
