Let’s determine the piecewise-defined functions for the graphs in problems 5 and 6. A piecewise-defined function is described by multiple expressions, each applying to a specific part of the domain.

Problem 5:

From the graph:

1. First segment (left part): A line from $x = -4$ to $x = -1$. The slope of this line is calculated using two points: $(-4, -2)$ and $(-1, 2)$.

   • Slope = $\frac{2 - (-2)}{-1 - (-4)} = \frac{4}{3}$
   • Equation of the line: $y = \frac{4}{3}x + \frac{10}{3}$

2. Second segment (middle part): A vertical line from $x = -1$ to $x = 0$. The value of $y$ is constant at $y = 2$.

3. Third segment (right part): A line starting at $x = 0$ and continuing to $x = 4$. The slope of this line is calculated using points $(0, 2)$ and $(4, 4)$.

   • Slope = $\frac{4 - 2}{4 - 0} = \frac{1}{2}$
   • Equation of the line: $y = \frac{1}{2}x + 2$

Piecewise Function for Problem 5: