38 / 84
2
Apply Compositions of Functions
In some real-world situations, functions are
applied in sequence, one after the other. In such cases, you can write a model for the
situation using a composition of functions.
SHOPPING
A new car dealer is discounting all new cars by 12%. At the same time,
the manufacturer is offering a $1500 rebate on all new cars. Mr. Navarro is buying a
car that is priced $24,500. Will the final price be lower if the discount is applied
before the rebate or if the rebate is applied before the discount?
Understand
Let
x
represent the original price of a new car,
d
(
x
) represent the price of
a car after the discount, and
r
(
x
) the price of the car after the rebate.
Plan
Write equations for
d
(
x
) and
r
(
x
).
The original price is discounted by 12%.
d
(
x
)
=
x
−
0.12
x
There is a $1500 rebate on all new cars.
r
(
x
)
=
x
−
1500
Solve
If the discount is applied
before
the rebate, then the final price of
Mr. Navarro’s new car is represented by [
r
◦
d
](24,500).
[
r
◦
d
](
x
)
=
r
[
d
(
x
)]
[
r
◦
d
](24,500)
=
r
[24,500
−
0.12(24,500)]
=
r
(24,500
−
2940)
=
r
(21,560)
=
21,560
−
1500
=
20,060
If the rebate is given
before
the discount is applied, then the final price of
Mr. Navarro’s car is represented by [
d
◦
r
](24,500).
[
d
◦
r
](
x
)
=
d
[
r
(
x
)]
[
d
◦
r
](24,500)
=
d
(24,500
−
1500)
=
d
(23,000)
=
23,000
−
0.12(23,000)
=
23,000
−
2760
=
20,240
[
r
◦
d
](24,500)
=
20,060 and [
d
◦
r
](24,500)
=
20,240. So, the final price of
the car is less when the discount is applied before the rebate.
Check
Graph [
d
◦
r
](
x
) and [
r
◦
d
](
x
). The final price of the car is less for [
r
◦
d
](
x
).
The answer seems reasonable because
C07-043A-888482-A
Final Price
20,500
20,000
21,000
19,500
0
Initial Price
24,750
24,500
24,250
25,000
[
d
◦
r
](
x
)
[
r
◦
d
](
x
)
the 12% discount is being applied to
a greater amount. Thus, the dollar
amount of the discount is greater.
Guided Practice
3.
SHOPPING
Gadgets-to-Go offers both an in-store
$300.
00
C06_001A_139256
$35 rebate and a 15% discount on a tablet that
normally sells for $300. Which provides
the better price: taking the discount before the
rebate or taking the discount after the rebate?
Real-World Example 3
Apply Compositions of Functions
Real-World Career
Adjusted for inflation, the
average price of a new car
increased from $22,013 in
2005 to about $30,000
in 2013.
Source:
Federal Trade Commission
Consumer Information
Kali9/iStock/Getty Images
324
|
Lesson 5-2
|
Composition of Functions