Answer
$$y=-\frac{1}{2} x+6$$
Work Step by Step
Since $f(x) $ has the tangent line $y=-2x+12 $ at $x=4$, then $f(x)$ passes through $$(4,-2(4)+12)=(4,4)$$
Since $g(x) $ is the inverse of $f(x)$, and $f, g$ are symmetric at $x=y$, then
\begin{align*}
g^{\prime}(4)&=\frac{1}{f^{\prime}(g(4))}\\
&=\frac{1}{f^{\prime}(4)}\\
&=\frac{1}{-2}
\end{align*}
Hence, the tangent line of $g(x)$ at $(4,4)$ is given by
\begin{align*}
\frac{y-y_1}{x-x_1}&=m\\
\frac{y-4}{x-4}&=\frac{-1}{2}
y&= -\frac{1}{2} x+6
\end{align*}
