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