1. **Problem Statement:** We are given a piecewise function $c(m)$ that represents cost based on miles driven $m$: $$ c(m) = \begin{cases} 100 & \text{if } 0 < m \leq 200 \\ 100 + 0.2m & \text{if } 200 < m \leq 500 \\ 150 + 0.1m & \text{if } m > 500 \end{cases} $$ We want to understand and graph this function. 2. **Understanding the function:** - For $0 < m \leq 200$, the cost is constant at 100. - For $200 < m \leq 500$, the cost increases linearly with slope 0.2 starting from $c(200) = 100 + 0.2 \times 200 = 140$. - For $m > 500$, the cost increases linearly with slope 0.1 starting from $c(500) = 150 + 0.1 \times 500 = 200$. 3. **Check continuity at boundaries:** - At $m=200$, from first piece $c(200) = 100$, from second piece $c(200) = 100 + 0.2 \times 200 = 140$. There is a jump, so the function is discontinuous here. - At $m=500$, from second piece $c(500) = 100 + 0.2 \times 500 = 200$, from third piece $c(500) = 150 + 0.1 \times 500 = 200$. The function is continuous here. 4. **Graph description:** - A horizontal line at $c(m) = 100$ from $m=0$ to $m=200$. - A line segment with slope 0.2 from $(200,140)$ to $(500,200)$. - A line segment with slope 0.1 starting at $(500,200)$ and increasing for $m > 500$. 5. **Summary:** This piecewise function models cost with a flat rate for the first 200 miles, a steeper increase for the next 300 miles, and a slower increase beyond 500 miles. This explanation helps understand the behavior and graph of $c(m)$ clearly.