Short Answer 
The piece-wise function g(x) has two segments: g(x) = 2x for x ‚a• 3 and g(x) = -1/3x + 7 for x ‚a§ 3. Key points (3, 6), (-6, 9), and (5, 10) are plotted, and lines are drawn to represent each segment accurately on the graph.
Step 1: Identify the Function Parts
The piece-wise function we are working with is given as: g(x) = {2x, x ‚a• 3; -frac{1}{3}x + 7, x ‚a§ 3}. Recognizing these two segments is crucial as they represent different behaviors based on the value of x. The first part applies when x is greater than or equal to 3, and the second part applies when x is less than or equal to 3. This segmentation helps in plotting the graph accurately.
Step 2: Plot Key Points
Identify and plot the key points that will help in graphing the function accurately. Start with the common point which is (3, 6), since this is where both parts of the function meet. Next, select one additional point from each segment to enhance the visual of the graph:
- For x ‚a§ 3: Plot (-6, 9).
- For x ‚a• 3: Plot (5, 10).
Step 3: Draw the Function Lines
Using the plotted points, we will now draw the two lines representing each segment of the piece-wise function. First, use the ray tool to connect the points (3, 6) and (5, 10) for the line of g(x) = 2x. Next, repeat this process for the segment g(x) = -frac{1}{3}x + 7 by connecting (3, 6) to (-6, 9). This will yield the complete graph of the piece-wise function.