Answer
The required values are $f\left( g\left( x \right) \right)=\frac{5x-56}{9}$ and $g\left( f\left( x \right) \right)=\frac{5x-4}{9}$. And the functions $f\left( x \right)=5x-9$ and $g\left( x \right)=\frac{x+5}{9}$ are not inverses of each other.
Work Step by Step
Consider the functions:
$f\left( x \right)=5x-9$
and
$g\left( x \right)=\frac{x+5}{9}$
The equation for $f$ is given as:
$f\left( x \right)=5x-9$
Replace $x$ with $g\left( x \right)$
$\begin{align}
& f\left( g\left( x \right) \right)=5g\left( x \right)-9 \\
& =5\left( \frac{x+5}{9} \right)-9 \\
& =\frac{5x+25-81}{9} \\
& =\frac{5x-56}{9}
\end{align}$
Now, to find $g\left( f\left( x \right) \right)$
Consider the function $g\left( x \right)$:
$g\left( x \right)=\frac{x+5}{9}$
Replace $x$ with $f\left( x \right)$
$\begin{align}
& g\left( f\left( x \right) \right)=\frac{f\left( x \right)+5}{9} \\
& =\frac{\left( 5x-9 \right)+5}{9} \\
& =\frac{5x-4}{9}
\end{align}$
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 $f\left( g\left( x \right) \right)=\frac{5x-56}{9}$ and $g\left( f\left( x \right) \right)=\frac{5x-4}{9}$. And the functions $f\left( x \right)=5x-9$ and $g\left( x \right)=\frac{x+5}{9}$ are not inverses of each other.
