Answer
$g(x)=-(x-2)^2+1$
Work Step by Step
The graph of $g$ is the result of shifting the graph of $f(x)=
x^2
$ by $2$ units to the right, reflecting about the $x-$axis, and shifting $1$ unit up. Hence, the equation of $g$ is
\begin{array}{l}\require{cancel}
g(x)=-(x-2)^2+1
.\end{array}