Answer
The required values are $f\left( g\left( x \right) \right)=x$ and $g\left( f\left( x \right) \right)=x$. And the functions $f\left( x \right)=3x+8$ and $g\left( x \right)=\frac{x-8}{3}$ are inverses of each other.
Work Step by Step
Consider the functions:
$f\left( x \right)=3x+8$
and
$g\left( x \right)=\frac{x-8}{3}$
The equation for $f$ is given as:
$f\left( x \right)=3x+8$
Replace $x$ with $g\left( x \right)$
$\begin{align}
& f\left( g\left( x \right) \right)=3g\left( x \right)+8 \\
& =3\left( \frac{x-8}{3} \right)+8 \\
& =x-8+8 \\
& =x
\end{align}$
Now, to find $g\left( f\left( x \right) \right)$
Consider the function $g\left( x \right)$:
$g\left( x \right)=\frac{x-8}{3}$
Replace $x$ with $f\left( x \right)$
$\begin{align}
& g\left( f\left( x \right) \right)=\frac{f\left( x \right)-8}{3} \\
& =\frac{\left( 3x+8 \right)-8}{3} \\
& =\frac{3x}{3} \\
& =x
\end{align}$
Because $g$ is inverse of $f$ (and vice-versa), the inverse notation can be used:
$f\left( x \right)=3x+8$ and $\text{ }{{f}^{-1}}\left( x \right)=\frac{x-8}{3}$
It can be easily observed that ${{f}^{-1}}$ can be expressed in $f$ by subtracting $8$ and dividing by $3$ .
$\begin{align}
& \text{ }{{f}^{-1}}\left( x \right)=\frac{x-8}{3} \\
& =g\left( x \right)
\end{align}$
Hence, the required values are $f\left( g\left( x \right) \right)=x$ and $g\left( f\left( x \right) \right)=x$. And the functions $f\left( x \right)=3x+8$ and $g\left( x \right)=\frac{x-8}{3}$ are inverses of each other.
