To express the function graphed in the image as a piecewise function, we need to analyze the different segments of the graph, their respective equations, and their domains.

Observations:

1. The graph consists of three distinct parts:
   • A line starting at $(-10, -1)$ and going up to $(-4, 6)$, where the point at $(-4, 6)$ is an open circle.
   • A second line starting from $(-4, 4)$ (solid circle) and going through the point $(-1, 9)$.
   • A single point at $(-5, -2)$, which is isolated and not part of the other lines.

Steps:

1. First line:

   • This line starts at $(-10, -1)$ and ends at $(-4, 6)$. Since it has an open circle at $(-4, 6)$, it means that the function is not defined at $x = -4$ on this segment.
   • The slope of the line is: $m = \frac{6 - (-1)}{-4 - (-10)} = \frac{7}{6}$ Using the point-slope form with point $(-10, -1)$: $y - (-1) = \frac{7}{6}(x - (-10))$ Simplifying: $y = \frac{7}{6}(x + 10) - 1$
   • Domain for this line: $-10 \leq x < -4$

2. Second line:

   • This line starts at $(-4, 4)$ and ends at $(-1, 9)$. It is solid at both endpoints, meaning it includes the points $(-4, 4)$ and $(-1, 9)$.
   • The slope of the line is: $m = \frac{9 - 4}{-1 - (-4)} = \frac{5}{3}$ Using point-slope form with $(-4, 4)$: $y - 4 = \frac{5}{3}(x - (-4))$ Simplifying: $y = \frac{5}{3}(x + 4) + 4$
   • Domain for this line: $-4 \leq x \leq -1$

3. Isolated point:

   • The point $(-5, -2)$ exists as a separate point, meaning $f(-5) = -2$.

Piecewise Function: