https://file.aiquickdraw.com/mathbot/file/172926579973035zu7vjo.jpg

Let's break down the problem from the image and develop the functions step by step.

### Step 1: Modeling the function with the coupon applied

The coupon gives Mr. Gluck **25% off** his entire order. Let $ x $ be the subtotal (the total amount before any discount or tax).

- Without a discount, the cost would be $ x $.
- After applying a 25% discount, Mr. Gluck will only pay 75% of the subtotal.

So, the function that models the cost after the coupon is:

\[
f(x) = 0.75x
\]

### Step 2: Adding Sales Tax

Next, the sales tax in Davidson County is **9.25%**. This means we need to add 9.25% of the cost after the discount.

- After the discount, the amount paid is $ 0.75x $.
- The sales tax adds $ 9.25\% $ to this, so the total cost becomes:

\[
f(x) = 0.75x \times 1.0925
\]

### Step 3: Combining the functions

To find the total amount Mr. Gluck will pay after both applying the coupon and the sales tax, we combine both the discount and the tax into one function. Simplifying:

\[
f(x) = 0.75x \times 1.0925 = 0.819375x
\]

Thus, the total cost function after applying both the coupon and the sales tax is:

\[
f(x) = 0.819375x
\]

This function represents how much Mr. Gluck will pay in total based on the subtotal of his order before applying the coupon.

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Let me know if you'd like further clarification or additional details.

Here are five relative questions to expand on this topic:
1. What would the total cost be if the coupon gave a 30% discount instead?
2. How would the function change if the sales tax were increased to 10%?
3. If Mr. Gluck spends $200, how much will he pay after the coupon and sales tax?
4. What would the total amount be without the coupon, only including the sales tax?
5. How could we use the derivative of the function to understand changes in total cost as the subtotal changes?

**Tip:** When dealing with percentage discounts and taxes, always remember to apply the discount before adding tax for accurate calculations!

Mr. Gluck is going to The Game Keep with a coupon for 25% off. Write a function modeling how much Mr. Gluck's purchase will be after applying the coupon, then account for a 9.25% sales tax. Combine and simplify the functions.

Learn how to calculate the total cost after applying a 25% discount and 9.25% sales tax. This solution models the process step by step, from applying the coupon to simplifying the final function.

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