Answer
Use symmetry about the line $y=x$
Work Step by Step
We are given the function:
$$g(x)=-x.$$
Graph the function $g$. Let $(x,y)$ be a point on the graph of $g$. Because $g(x)=-x$, the coordinates can be written as $(x,-x)$. The symmetric of this point with respect to the line $y=x$ belongs to the graph of $g^{-1}$. But this symmetric is $(-x,x)$ which belongs to the graph of $g$. Therefore the two graph are identical, so $g(x)=g^{-1}(x)$.
Find the inverse $g^{-1}$ of the function $g$ algebraically:
$$\begin{align*}
g(x)&=-x\quad&&\text{Write original function.}\\
y&=-x\quad&&\text{Replace }g^{-1}(x)\text{ by }y\\
x&=-y\quad&&\text{Switch }x\text{ and }y.\\
-x&=y\quad&&\text{Divide each side by }-1.
\end{align*}$$
The inverse function of $g$ is $g^{-1}(x)=-x$.
