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The question involves choosing which coupon Tara should apply first to maximize the discount on an item priced above $20. Let's break down the situation and the mathematical logic involved.

### Information Given:
- Tara has two coupons:
  1. **Coupon 1:** Gives **$10 off** on one item.
  2. **Coupon 2:** Gives **10% off** on one item.
  
- Tara can apply both coupons, but the order in which she applies them can change the final price.
  
- Let $ f(x) $ be the function representing the price after applying **Coupon 1** (flat $10 off), and let $ g(x) $ be the function representing the price after applying **Coupon 2** (10% off).
  
- Tara's goal is to get the **lowest price**, and the original price of the item is more than $20.

### Step-by-Step Analysis:

#### Functions:
- If $ x $ is the original price:
  - **Applying Coupon 1 (flat $10 off):**
    \[
    f(x) = x - 10
    \]
  - **Applying Coupon 2 (10% off):**
    \[
    g(x) = 0.9x
    \]

#### Case 1: Apply Coupon 1 before Coupon 2
1. First, apply Coupon 1:
   \[
   f(x) = x - 10
   \]
2. Then apply Coupon 2 to the result:
   \[
   g(f(x)) = 0.9(x - 10)
   \]
   Simplified:
   \[
   g(f(x)) = 0.9x - 9
   \]

#### Case 2: Apply Coupon 2 before Coupon 1
1. First, apply Coupon 2:
   \[
   g(x) = 0.9x
   \]
2. Then apply Coupon 1 to the result:
   \[
   f(g(x)) = 0.9x - 10
   \]

#### Comparison of Results:
- If you apply **Coupon 1 before Coupon 2**, the final price is:
  \[
  0.9x - 9
  \]
- If you apply **Coupon 2 before Coupon 1**, the final price is:
  \[
  0.9x - 10
  \]
  
Since $ 0.9x - 10 $ is always **less** than $ 0.9x - 9 $ for $ x > 20 $, Tara will get a lower price by **applying Coupon 2 first**, then applying Coupon 1.

### Conclusion:
The correct choice is **Option D**, as applying **Coupon 2 before Coupon 1** gives Tara a lower price.

Let me know if you'd like any further details or clarifications! Here are five related questions to help deepen understanding:

1. How would the answer change if the item’s price were less than $20?
2. What if Coupon 1 offered a percentage off instead of a flat discount?
3. How would the order of applying coupons affect the result if both coupons were percentage-based?
4. Can you derive a general formula for combining different types of discounts?
5. How does the concept of order of operations apply in this real-world problem?

**Tip:** When working with multiple discounts, apply percentage-based discounts first to get the maximum savings.

Tara has two store coupons. Coupon 1 gives her $10 off on one item. Coupon 2 gives her 10% off on one item. Tara can use both coupons together on one item in the store and choose which coupon to apply first. For an item that has an original price of x dollars, let f be the function that models the discount price of the item using only coupon 1 and let g be the function that models the discount price of the item using only coupon 2. The price of each item in this store is more than $20. Which of the following statements best explains which coupon should be applied first when using both coupons to give Tara a lower price?

Learn how to determine the optimal coupon application order between a $10 off coupon and a 10% off coupon using functions. Tara needs to choose which coupon to apply first to minimize the final price. Suitable for students in Grades 8-10.

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