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Let's solve each part of the problem step-by-step:

### Problem Analysis:
- You have a $40 coupon and a 20% discount available for purchasing a cell phone.
- $ x $ represents the regular price of the cell phone.

### (a) Find the function $ f $ when only the 20% discount applies:
The 20% discount means you pay 80% of the regular price.

\[
f(x) = 0.8x
\]

### (b) Find the function $ g $ when only the $40 coupon applies:
The $40 coupon reduces the price by a fixed amount of $40.

\[
g(x) = x - 40
\]

### (c) Find the compositions $ (f \circ g)(x) $ and $ (g \circ f)(x) $:

1. **Composition $ (f \circ g)(x) $:**
   This means applying the $40 coupon first, then the 20% discount.

   \[
   (f \circ g)(x) = f(g(x)) = f(x - 40) = 0.8(x - 40)
   \]

   Simplifying this:

   \[
   (f \circ g)(x) = 0.8x - 32
   \]

2. **Composition $ (g \circ f)(x) $:**
   This means applying the 20% discount first, then the $40 coupon.

   \[
   (g \circ f)(x) = g(f(x)) = g(0.8x) = 0.8x - 40
   \]

### Determine which composition gives the lower price:
- $ (f \circ g)(x) = 0.8x - 32 $
- $ (g \circ f)(x) = 0.8x - 40 $

Since $ -40 $ is less than $ -32 $, the composition $ (g \circ f)(x) $ gives the lower price.

**Answer:**
Applying the discount followed by the coupon gives the lower price.

---

Do you have any questions or would you like more details? Here are some related questions to explore:

1. How does changing the discount percentage affect the results?
2. What if the coupon amount is different, like $50 instead of $40?
3. How would the compositions change if both discounts were applied simultaneously?
4. What is the effect of reversing the order of application for different types of discounts (e.g., percentage vs. fixed amount)?
5. How do you determine the order of operations when combining multiple discounts?

**Tip:** Always apply the percentage discount first when possible, as it reduces the higher initial price, maximizing the savings.

You have a $40 coupon from the manufacturer that is good for the purchase of a cell phone. The store where you are purchasing your cell phone is offering a 20% discount on all cell phones. Let x represent the regular price of the cell phone. (a) Suppose only the 20% discount applies. Find a function f that models the purchase price of the cell phone as a function of the regular price x. (b) Suppose only the $40 coupon applies. Find a function g that models the purchase price of the cell phone as a function of the sticker price x. (c) If you can use the coupon and the discount, then the purchase price is either (f ∘ g)(x) or (g ∘ f)(x), depending on the order in which they are applied to the price. Find both (f ∘ g)(x) and (g ∘ f)(x). Which composition gives the lower price?

Learn how to model the purchase price of a cell phone using function composition when both a 20% discount and a $40 coupon are available. Find out which order of applying the discount and coupon gives the lower price. Suitable for Grades 9-11 students learning about functions and algebra.

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