Sales A car dealer offers a 10% discount off the list price...

Sales A car dealer offers a 10$\%$ discount off the list price $x$ for any car on the lot. At the same time, the manufacturer offers a $\$ 2000$ rebate for each purchase of a car. a. Write a function $f(x)$ to represent the price after the discount. b. Write a function $g(x)$ to represent the price after the $\$ 2000$ rebate. c. Suppose the list price of a car is $\$ 18,000$ . Use a composite function to find the price of the car if the discount is applied before the rebate. d. Suppose the list price of a car is $\$ 18,000$ . Use a composite function to find the price of the car if the rebate is applied before the discount.

Sales A car dealer offers a 10$\%$ discount off the list price $x$ for any car on the lot. At the same time, the manufacturer offers a $\$ 2000$ rebate for each purchase of a car.
a. Write a function $f(x)$ to represent the price after the discount.
b. Write a function $g(x)$ to represent the price after the $\$ 2000$ rebate.
c. Suppose the list price of a car is $\$ 18,000$ . Use a composite function to find the price of the car if the discount is applied before the rebate.
d. Suppose the list price of a car is $\$ 18,000$ . Use a composite function to find the price of the car if the rebate is applied before the discount.

Key Concepts

Linear Functions

Linear functions describe relationships where the change in the output is proportional to the change in the input. In price calculations, these functions emerge naturally when applying constant percentage discounts or fixed rebates, leading to operations that add or subtract values in a uniform way regardless of the initial amount.

Percentage Discounts

Percentage discounts reduce a price by a fixed percentage of the original amount. This concept is used to calculate the new price by multiplying the original price by a factor (e.g., 0.90 for a 10% discount), representing a proportional decrease in the price.

Fixed Rebates

Fixed rebates subtract a constant amount from the price regardless of its value. This is modeled by a linear function where a constant quantity is deducted from the already calculated price, leading to a straightforward reduction due to the rebate amount.

Composite Functions

Composite functions involve applying one function to the result of another. This concept is key when calculating the final price after sequential operations, such as applying a discount first and then a rebate, or vice versa, to accurately model multi-step price adjustments.

Order of Operations in Composite Functions

The order in which functions are applied in composition matters, as applying a discount before a rebate can lead to a different result compared to applying the rebate first and then the discount. Understanding this concept is crucial for determining accurate outcomes in multi-step financial calculations.

Transcript

00:02 In this problem, it says that the dealership will give you 10 % discount on your new car.

00:18 But the manufacturer will give you a 2 ,000 discount.

00:38 And then x equals the amount that you paid for your car.

00:48 So it wants us to find three things.

00:50 A, it wants you to make an equation for the discount of your dealership.

00:55 So this is your dealership discount.

01:04 So my dealership discount, first i have to take it to account my cost of my car.

01:10 So this would be the cost of my car, which is x minus my discount.

01:20 My discount is 10 % of my car.

01:23 10 % changed into a decimal.

01:28 I moved my decimal over two points.

01:33 It will be 0 .1 times how much i paid for the car.

01:42 So this f of x is the amount that i would pay for the car if i got the dealership discount.

01:51 Now let's look at what i would pay for the car if i had the manufacturer discount.

02:04 That's going to be g of x.

02:10 So the manufacturer would be what i paid for the car minus just the $2 ,000.

02:25 Now we're going to compare what would be a better deal, you getting the dealer.

02:33 Up dealership discount first, then your manufacturer discount, or the manufacturer discount, then the dealership discount.

02:42 So in order to do that first, we're going to find out what is g of f.

02:47 G of f is when i get the dealership discount first and then the manufacturer discount.

02:53 So i'm going to replace all the x's in g of x with my f of x.

03:01 So that looks like x minus 1 .0 .1 x minus 2 ,000...

Please give Ace some feedback