A new store opens in New York. You received a coupon for $5.00 off a pair of jeans j(x) = x-5. When you get to the store you learn that they are giving a 25% discount d(x)=.75x
a) Write a composite function for the cost of the jeans if the 25% discount is applied after the $5.00 coupon
b) Write a composite function for the cost of the jeans if the 25% discount is applied before the $5.00 coupon
c) Using both composite functions determine what you will pay for a pair of jeans with a regular price of $30.00? Which is a better deal?
(a) d(j(x)) = .75(j(x)) = .75(x-5)
(b) j(d(x)) = d(x)-5 = .75x - 5
(c) go for it
To answer these questions, we will need to create composite functions and evaluate them using the given conditions.
a) To write a composite function for the cost of the jeans when the 25% discount is applied after the $5.00 coupon, we need to first calculate the cost of the jeans after the coupon is applied, and then apply the 25% discount on the resulting price.
Let's break it down step by step:
1. Apply the $5.00 coupon to the original price: j(x) = x - 5
(This represents the cost of the jeans after the coupon is applied)
2. Apply the 25% discount to the resulting price from step 1:
d(j(x)) = 0.75 * j(x)
= 0.75 * (x - 5)
= 0.75x - 3.75
Therefore, the composite function for the cost of the jeans with the 25% discount applied after the $5.00 coupon is d(j(x)) = 0.75x - 3.75.
b) To write a composite function for the cost of the jeans when the 25% discount is applied before the $5.00 coupon, we need to first apply the 25% discount to the original price, and then subtract the $5.00 coupon from the resulting price.
Again, let's break it down step by step:
1. Apply the 25% discount to the original price: d(x) = 0.75x
(This represents the cost of the jeans after the discount is applied)
2. Apply the $5.00 coupon to the resulting price from step 1:
j(d(x)) = d(x) - 5
= 0.75x - 5
Therefore, the composite function for the cost of the jeans with the 25% discount applied before the $5.00 coupon is j(d(x)) = 0.75x - 5.
c) Now, we can use both composite functions to determine the cost of a pair of jeans with a regular price of $30.00 and compare which option is a better deal (lower price).
Using the first composite function (25% discount after $5.00 coupon):
d(j(30)) = 0.75 * (30 - 5)
= 0.75 * 25
= $18.75
Using the second composite function (25% discount before $5.00 coupon):
j(d(30)) = 0.75 * 30 - 5
= 22.5 - 5
= $17.50
Therefore, with a regular price of $30.00, the cost of the jeans would be $18.75 with the 25% discount applied after the $5.00 coupon, and $17.50 with the 25% discount applied before the $5.00 coupon. The second option, applying the discount before the coupon, is a better deal as it results in a lower price.