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 $f\left( x \right)=\frac{2}{x-5}$ and $g\left( x \right)=\frac{2}{x}+5$ are inverses of each other.
Work Step by Step
Consider the functions:
$f\left( x \right)=\frac{2}{x-5}$
and
$g\left( x \right)=\frac{2}{x}+5$
The equation for $f$ is given as:
$f\left( x \right)=\frac{2}{x-5}$
Replace $x$ with $g\left( x \right)$
$\begin{align}
& f\left( g\left( x \right) \right)=\frac{2}{g\left( x \right)-5} \\
& =\frac{2}{\left( \frac{2}{x}+5 \right)-5} \\
& =\frac{2}{\frac{2}{x}} \\
& =x
\end{align}$
Now, to find $g\left( f\left( x \right) \right)$
Consider the function $g\left( x \right)$:
$g\left( x \right)=\frac{2}{x}+5$
Replace $x$ with $f\left( x \right)$
$\begin{align}
& g\left( f\left( x \right) \right)=\frac{2}{f\left( x \right)}+5 \\
& =\frac{2}{\frac{2}{x-5}}+5 \\
& =\frac{2\left( x-5 \right)}{2}+5 \\
& =x
\end{align}$
Because $g$ is inverse of $f$ (and vice-versa), the inverse notation can be used:
$f\left( x \right)=\frac{2}{x-5}$ and $\text{ }{{f}^{-1}}\left( x \right)=\frac{2}{x}+5$
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 $f\left( x \right)=\frac{2}{x-5}$ and $g\left( x \right)=\frac{2}{x}+5$ are inverses of each other.
