A real estate broker charges a commission of 6% on Sales valued up to$300,000. For sales valued at
more than$ 300,000, the commission is $ 6,000 plus 4% of the sales price.
a. Represent the commission earned as a function R.
b. Find R (200,000).
c. Find R (500,000).

### 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:
\[
R(s) =
\begin{cases}
0.06s & \text{if } s \leq 300,000 \\
6,000 + 0.04(s - 300,000) & \text{if } s > 300,000
\end{cases}
\]

### 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?

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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.

A real estate broker charges a commission of 6% on sales valued up to $300,000. For sales valued at more than $300,000, the commission is $6,000 plus 4% of the sales price. a. Represent the commission earned as a function R. b. Find R(200,000). c. Find R(500,000).

Learn how to calculate real estate commission using piecewise functions. This problem involves determining the commission for sales below and above $300,000, including a 6% rate for lower sales and a 4% rate for higher sales with an additional fixed amount. Explore the function representation and evaluate specific cases for $200,000 and $500,000.

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