To determine the piecewise function $f(x)$ based on the graph, I will analyze the two line segments and their corresponding domains. Each line segment represents a portion of the function.

1. Left Line Segment (From $x = -5$ to $x = 1$):

• Endpoints: The line starts at $(-5, 6)$ and ends at $(1, -2)$.
• Slope Calculation: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 6}{1 - (-5)} = \frac{-8}{6} = -\frac{4}{3}$
• Equation of the Line: Use the point-slope form $y - y_1 = m(x - x_1)$ with $m = -\frac{4}{3}$ and point $(-5, 6)$: $y - 6 = -\frac{4}{3}(x + 5)$ Simplifying this: $y - 6 = -\frac{4}{3}x - \frac{20}{3}$ $y = -\frac{4}{3}x + \frac{-20 + 18}{3} = -\frac{4}{3}x - \frac{2}{3}$ Thus, for $-5 \leq x \leq 1$, the equation is: $f(x) = -\frac{4}{3}x - \frac{2}{3}$

2. Right Line Segment (From $x = 1$ to $x = 5$):

• Endpoints: The line starts at $(1, -2)$ and ends at $(5, 4)$.
• Slope Calculation: $\text{slope} = \frac{4 - (-2)}{5 - 1} = \frac{6}{4} = \frac{3}{2}$
• Equation of the Line: Use the point-slope form $y - y_1 = m(x - x_1)$ with $m = \frac{3}{2}$ and point $(1, -2)$: $y + 2 = \frac{3}{2}(x - 1)$ Simplifying this: $y + 2 = \frac{3}{2}x - \frac{3}{2}$ $y = \frac{3}{2}x - \frac{3}{2} - 2 = \frac{3}{2}x - \frac{7}{2}$ Thus, for $1 < x \leq 5$, the equation is: $f(x) = \frac{3}{2}x - \frac{7}{2}$

Final Piecewise Function:

The function $f(x)$ is given by: