Part a: Represent the commission earned as a function $R(s)$

Let $s$ represent the sales price in dollars.

The commission structure can be described in two cases:

1. For sales up to $300,000: $R(s) = 0.06s \quad \text{for } s \leq 300,000$

2. For sales over $300,000: $R(s) = 6,000 + 0.04(s - 300,000) \quad \text{for } s > 300,000$

Thus, the function $R(s)$ can be written as:

\begin{cases} 0.06s & \text{if } s \leq 300,000 \\ 6,000 + 0.04(s - 300,000) & \text{if } s > 300,000 \end{cases}$$ ### Part b: Find $$R(200,000)$$ Since $$200,000 \leq 300,000$$, we use the first case of the function: $$R(200,000) = 0.06 \times 200,000 = 12,000$$ ### Part c: Find $$R(500,000)$$ Since $$500,000 > 300,000$$, we use the second case of the function: $$R(500,000) = 6,000 + 0.04 \times (500,000 - 300,000)$$ $$R(500,000) = 6,000 + 0.04 \times 200,000 = 6,000 + 8,000 = 14,000$$ ### Summary of Results: - $$R(200,000) = 12,000$$ - $$R(500,000) = 14,000$$ Do you need any more details or further clarification? --- Here are 5 questions to further explore this topic: 1. How would the commission change if the rate for sales over $300,000 were different? 2. What is the total commission for a sale of $350,000? 3. How does the commission structure affect sales near the $300,000 mark? 4. What is the break-even point where the two commission rates (before and after $300,000) meet? 5. How would the function change if a flat fee was added to the commission for all sales? **Tip:** Understanding piecewise functions helps in modeling real-world scenarios where different conditions apply based on thresholds.