Answer
The required values are $f\left( g\left( x \right) \right)=\frac{3x-40}{7}$ and $g\left( f\left( x \right) \right)=\frac{3x-4}{7}$. And the functions $f\left( x \right)=3x-7$ and $g\left( x \right)=\frac{x+3}{7}$ are not inverses of each other.
Work Step by Step
Consider the functions:
$f\left( x \right)=3x-7$
and
$g\left( x \right)=\frac{x+3}{7}$
The equation for $f$ is given as:
$f\left( x \right)=3x-7$
Replace $x$ with $g\left( x \right)$
$\begin{align}
& f\left( g\left( x \right) \right)=3g\left( x \right)-7 \\
& =3\left( \frac{x+3}{7} \right)-7 \\
& =\frac{3x+9-49}{7} \\
& =\frac{3x-40}{7}
\end{align}$
Now, to find $g\left( f\left( x \right) \right)$
Consider the function $g\left( x \right)$:
$g\left( x \right)=\frac{x+3}{7}$
Replace $x$ with $f\left( x \right)$
$\begin{align}
& g\left( f\left( x \right) \right)=\frac{f\left( x \right)+3}{7} \\
& =\frac{\left( 3x-7 \right)+3}{7} \\
& =\frac{3x-4}{7}
\end{align}$
Thus, $f\left( g\left( x \right) \right)\ne x$ and $g\left( f\left( x \right) \right)\ne x$
Therefore, the required values are $f\left( g\left( x \right) \right)=\frac{3x-40}{7}$ and $g\left( f\left( x \right) \right)=\frac{3x-4}{7}$. And the functions $f\left( x \right)=3x-7$ and $g\left( x \right)=\frac{x+3}{7}$ are not inverses of each other.
