The graph in the image represents a piecewise-defined function with different segments. Let's break down the function based on the different sections of the graph.

1. For $x < -4$:

The graph has a constant value for $x < -4$. Observing the graph, the value of $y$ remains constant at $-3$ in this interval. So, the function for this section is: $f(x) = -3 \quad \text{for} \quad x < -4$

2. For $-4 \leq x \leq 4$:

In this interval, the graph forms a straight line with a positive slope. We need to determine the equation of this line. To do so, we can use the slope-intercept form, $y = mx + b$, where $m$ is the slope, and $b$ is the y-intercept.

• Find the slope (m): The slope is given by: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Picking two points from the line: $(-4, -3)$ and $(4, 5)$: $m = \frac{5 - (-3)}{4 - (-4)} = \frac{8}{8} = 1$ So, the slope $m = 1$.

• Find the y-intercept (b): Now that we know the slope, we can substitute one of the points, say $(-4, -3)$, into the equation $y = mx + b$ to find the intercept $b$: $-3 = 1(-4) + b \quad \Rightarrow \quad b = 1$ Therefore, the equation of the line is: $f(x) = x + 1 \quad \text{for} \quad -4 \leq x \leq 4$

3. For $x > 4$:

For $x > 4$, the graph shows a constant value of $5$. Therefore, the function for this interval is: $f(x) = 5 \quad \text{for} \quad x > 4$

Final Piecewise Function: