Mrs. Spray and Dr. H decide to open a bakery together. Don’t worry they will still teach
Precalculus! They are having a sale on chocolate nut brownies and brownies are on
sale for 15%. You earned an A on your last Precalculus test so you also have a $2 off
coupon. Let x be the original price of the brownies.
a) Write a function to represent the cost of the brownies after the discount sale.
b) Write a function to represent the cost of the brownies after the coupon.
c) Write a composite function that represents the cost of the brownies if you first use
the coupon and then get a discount.
d) Write a composite function that represents the cost of the brownies if you first get
the discount and then use the coupon.
e) Which has the lower price: using the coupon first and then the discount or using the
discount first and then the coupon.
To answer these questions, we need to break down the problem and identify the steps involved.
1) Write a function to represent the cost of the brownies after the discount sale.
Since the brownies are on sale for 15% off, we can write the function as:
f(x) = x - 0.15x
= 0.85x
So, the cost of the brownies after the discount sale is 0.85 times the original price.
2) Write a function to represent the cost of the brownies after the coupon.
Since you have a $2 off coupon, the function can be written as:
g(x) = x - $2
So, the cost of the brownies after applying the coupon is the original price minus $2.
3) Write a composite function that represents the cost of the brownies if you first use the coupon and then get a discount.
To find the composite function, we need to apply g(x) first and then f(x).
h(x) = f(g(x))
= f(x - $2)
= 0.85(x - $2)
So, the cost of the brownies after applying the coupon and then the discount is 0.85 times the original price minus $1.70 (0.85 times $2).
4) Write a composite function that represents the cost of the brownies if you first get the discount and then use the coupon.
To find the composite function, we need to apply f(x) first and then g(x).
k(x) = g(f(x))
= g(0.85x)
= 0.85x - $2
So, the cost of the brownies after applying the discount and then the coupon is 0.85 times the original price minus $2.
5) To determine which has the lower price: using the coupon first and then the discount or using the discount first and then the coupon, we need to compare h(x) and k(x).
Using the coupon first and then the discount (h(x)) gives us a cost of 0.85 times the original price minus $1.70.
Using the discount first and then the coupon (k(x)) gives us a cost of 0.85 times the original price minus $2.
Comparing the two, we can see that h(x) has a lower price than k(x) since $1.70 is less than $2.
Therefore, using the coupon first and then the discount will result in a lower price for the brownies.