To represent the given graph as a piecewise function, we need to identify the equation of each linear segment and the domain over which each segment is defined.

1. First Segment:

• Line Segment: Passes through the points $(-2, 2)$ and $(-1, 3)$.

• Equation of the line:

  Find the slope ($m$) of the line: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 2}{-1 - (-2)} = \frac{1}{1} = 1$

  Using the point-slope form: $y - y_1 = m(x - x_1)$ Substitute $(x_1, y_1) = (-2, 2)$ and $m = 1$: $y - 2 = 1(x + 2)$ Simplifying: $y = x + 4$

  Domain: $-2 \leq x \leq -1$

2. Second Segment:

• Line Segment: Passes through the points $(-1, 0)$ and $(3, -1)$.

• Equation of the line:

  Find the slope ($m$) of the line: $m = \frac{-1 - 0}{3 - (-1)} = \frac{-1}{4} = -\frac{1}{4}$

  Using the point-slope form: $y - 0 = -\frac{1}{4}(x + 1)$ Simplifying: $y = -\frac{1}{4}x - \frac{1}{4}$

  Domain: $-1 < x < 3$ (Note that $(-1, 0)$ and $(3, -1)$ are not included)

3. Third Segment:

• Line Segment: Passes through the points $(3, -3)$ and $(4, -3)$.

• Equation of the line:

  The line is horizontal, so the equation is: $y = -3$

  Domain: $3 \leq x \leq 4$

Piecewise Function:

Combining all the segments, the piecewise function is:

x + 4 & \text{if } -2 \leq x \leq -1 \\ -\frac{1}{4}x - \frac{1}{4} & \text{if } -1 < x < 3 \\ -3 & \text{if } 3 \leq x \leq 4 \end{cases}$$ Would you like more details or have any questions? --- **Related Questions:** 1. How do you find the equation of a line given two points? 2. What are the steps to determine the slope of a line? 3. How do you represent a horizontal line as a function? 4. What does an open dot versus a closed dot signify on a graph? 5. How do you determine the domain for each piece of a piecewise function? **Tip:** When working with piecewise functions, carefully check each interval to ensure that it correctly represents the function's graph.