Answer
The function is $g\left( x \right)=-\left| x-5 \right|+1.$
Work Step by Step
The graph of $g\left( x \right)$ is obtained by doing a transformation in the graph of $f\left( x \right)=\left| x \right|.$
The vertex of the function $f\left( x \right)=\left| x \right|$ is at the origin. The vertex of the function $g\left( x \right)$ is at $x=5$, so the graph of $f\left( x \right)=\left| x \right|$ will be shifted 5 units towards the right.
$f\left( x \right)=\left| x \right|$ , $c=5.$
Then, the function becomes
$\begin{align}
& h\left( x \right)=\left| x-c \right| \\
& =\left| x-5 \right|
\end{align}$
The graph of $g\left( x \right)$ is a shifted reflection of the graph of $f\left( x \right)=\left| x \right|$ along the x axis. Thus, the coordinates of the $y\text{-}$ axis are multiplied by $-1$.
So, the function becomes
$\begin{align}
& h'\left( x \right)=-h\left( x \right) \\
& =-\left| x-5 \right|
\end{align}$
Now, the transformed graph is shifted vertically upward by 1 unit; thus, the y-coordinates of all the points of the graph are increased by 1 unit. Thus, add 1 to the function.
Therefore, the final function will become
$\begin{align}
& g\left( x \right)=h'\left( x \right)+1 \\
& =-\left| x-5 \right|+1
\end{align}$
Hence, the function is $g\left( x \right)=-\left| x-5 \right|+1$.
