This document discusses Carlton purchasing a $50 pair of jeans using different discount options: 1) 20% off store promotion, 2) $15 customer reward coupon, 3) 20% off promotion then $15 coupon, 4) $15 coupon then 20% off promotion. It determines that the order discounts are applied matters, with the 20% then $15 coupon being the best deal at $25, while the $15 coupon then 20% off is $28. It relates this real world example to the concept of function composition, where each discount represents a function and the order changes the overall function.

1. Composition of Functions!
A REAL WORLD EXAMPLE

2. Let’s face it! Most Americans enjoy a great pair of jeans. And buying them on sale
seems like an awesome way to get more bang for your buck!

3. In our application we will look at the effect of two different deductions and
discounts, and determine which combination of deductions is better for the
consumer.

4. OUR ASSIGNMENT IS TO DETERMINE CARLTON’S COST TO PURCHASE A PAIR OF JEANS
WHEN HE PURCHASES A $50 PAIR OF JEANS BY :
a) USING ONLY THE STORE PROMOTION OF 20% OFF
b) USING ONLY THE $15 CUSTOMER REWARD COUPON
c) USING THE STORE PROMOTION OF 20% OFF AND THEN THE $15 COUPON
d) USING THE $15 OFF COUPON AND THEN THE STORE PROMOTION OF 20%

5. A) WHAT WILL CARLTON’S COST BE TO PURCHASE A $50 PAIR OF
JEANS IF HE USES ONLY THE 20% OFF STORE PROMOTION?
$𝟓𝟎 . 𝟐𝟎 = $𝟏𝟎
$𝟓𝟎 − $𝟏𝟎 = $𝟒𝟎

6. B) WHAT WILL CARLTON’S COST BE TO PURCHASE A $50 PAIR OF JEANS IF
HE USES ONLY THE $15 OFF CUSTOMER LOYALTY COUPON?
$𝟓𝟎 − $𝟏𝟓 = $𝟑𝟓

7. C) WHAT WILL CARLTON’S COST BE TO PURCHASE A $50 PAIR OF JEANS IF
HE USES THE 20% OFF STORE DISCOUNT AND THEN THE $15 COUPON?
ORIGINAL PRICE – 20% DISCOUNT – COUPON = COST
$𝟓𝟎 − 𝟓𝟎 . 𝟐𝟎 − 𝟏𝟓 = $𝟐𝟓

8. D) WHAT WILL CARLTON’S COST BE TO PURCHASE A $50 PAIR OF JEANS IF
HE USES THE $15 COUPON AND THEN THE 20% OFF STORE DISCOUNT?
ORIGINAL PRICE – COUPON – 20%
0FF THE REDUCED PRICE = COST
$𝟓𝟎 − $𝟏𝟓 − $𝟑𝟓 . 𝟐𝟎 =
$𝟓𝟎 − $𝟏𝟓 − $𝟕 = $𝟐𝟖

9. It DOES matter the order that the
discounts are applied.
How does this apply to what we will learn about composition of functions?
- Each discount is a function of the cost of the jeans. In one function we are reducing
the cost by 20%. In the other function we are subtracting $15 from the cost.
- When we use BOTH of these functions, the order that we apply them makes a
difference in the answer.

10. Composition of Functions – The BASICS
𝑓 𝑥 = 𝑥 − .20 𝑥
𝑔 𝑥 = 𝑥 − 15
This is a function of 20% off a price.
This is a function of $15 off a price.
𝑔 𝑓 𝑥 represents applying the 20% off discount first. Notice how the innermost function is the % off.
𝑓 𝑔 𝑥 represents applying the coupon first and then the 20% reduction. Notice how the inner most function
is the coupon.

11. Composition of Functions – The BASICS
𝑓 𝑥 = 𝑥 − .20 𝑥
𝑔 𝑥 = 𝑥 − 15
This is a function of 20% off a price.
This is a function of $15 off a price.
𝑔 𝑓 𝑥 = 𝑥 − .20 𝑥 − 15
𝑔 𝑓 𝑥 = .80𝑥 − 15
𝑓 𝑔 𝑥 = 𝑥 − 15 − .20 𝑥 − 15
𝑓 𝑔 𝑥 = 𝑥 − 15 − .20𝑥 + 3
𝑓 𝑔 𝑥 = .80𝑥 − 12