To determine the value of $x$ for which $g(x) = -2$, we need to find the point on the graph where the function $g(x)$ intersects the line $y = -2$.

Upon analyzing the graph, we observe the following:

• The function $g(x)$ intersects the line $y = -2$ at the point where the x-coordinate is $-2$.

Therefore, the value of $x$ for which $g(x) = -2$ is $\boxed{-2}$.

Would you like any further details or have any questions?

Here are some related questions you might find interesting:

1. What is the slope of the line segment from $x = -6$ to $x = -2$?
2. What is the slope of the line segment from $x = -2$ to $x = 3$?
3. What are the coordinates of the point where $g(x)$ intersects the y-axis?
4. What is the value of $g(x)$ when $x = 3$?
5. For what values of $x$ is $g(x) = 0$?
6. What is the domain of the function $g(x)$?
7. What is the range of the function $g(x)$?
8. What are the x-intercepts and y-intercepts of the function $g(x)$?

Tip: When analyzing piecewise linear functions on a graph, pay attention to the points where the slope changes and note these critical points for a complete understanding of the function's behavior.