From the image, we are tasked with completing the piecewise function based on the graph provided. Here's how to approach it:

Step-by-Step Breakdown:

1. Identify the first piece (from $x = -6$ to $x = -3$):

• The graph shows a horizontal line from $x = -6$ to $x = -3$ at $y = -2$.
• So, the first piece of the function is: $f(x) = -2 \quad \text{if } -6 \leq x \leq -3$

2. Identify the second piece (from $x = -3$ to $x = 2$):

• There is a line segment between $(-3, 1)$ and $(2, 3)$.
• To find the equation of this line, use the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$: $m = \frac{3 - 1}{2 - (-3)} = \frac{2}{5}$ The point-slope form is $y - y_1 = m(x - x_1)$, using the point $(-3, 1)$: $y - 1 = \frac{2}{5}(x + 3)$ Simplifying: $y = \frac{2}{5}x + \frac{6}{5} + 1 = \frac{2}{5}x + \frac{11}{5}$ So, the second piece of the function is: $f(x) = \frac{2}{5}x + \frac{11}{5} \quad \text{if } -3 < x \leq 2$

3. Identify the third piece (from $x = 2$ to $x = 6$):

• This part is already given: $f(x) = 2x - 7$ for $2 < x \leq 6$.

Final Piecewise Function: