Determine the function graphed in the image.

Solution:

The image displays a piecewise function and its graph. We need to determine the function definitions for each part.

The function is defined as:

• \( y = -\frac{1}{2}x + 3 \) for \( x \ge 2 \)
• \( y = x - 1 \) for \( x < 2 \)

This can be represented as:

Let's verify the points on the graph:

• For \( x = 2 \), the first part of the function gives \( y = -\frac{1}{2}(2) + 3 = -1 + 3 = 2 \). The point \( (2, 2) \) is plotted.
• For \( x = 4 \), the first part gives \( y = -\frac{1}{2}(4) + 3 = -2 + 3 = 1 \). The point \( (4, 1) \) is plotted.
• For \( x = 0 \), the second part gives \( y = 0 - 1 = -1 \). The point \( (0, -1) \) is plotted.
• For \( x = 1 \), the second part gives \( y = 1 - 1 = 0 \). The point \( (1, 0) \) is plotted.

The tables provided also represent these functions:

The graph visually confirms the behavior of these two linear functions meeting at \( x=2 \).

Answer: The function is defined by \( y = -\frac{1}{2}x + 3 \) for \( x \ge 2 \) and \( y = x - 1 \) for \( x < 2 \).