Problem 43 - Algebra 2

Question

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.

Step-by-Step Solution

Answer

a. The price after the discount is represented by the function \(f(x) = 0.90x\). b. The price after the rebate is represented by the function \(g(x) = x - 2000\). c. If the discount is applied before the rebate, the price of car is $14,200. d. If the rebate is applied before the discount, the price of car is $14,400.

Step 1: Define the discount function

The discount function is defined as the list price minus the % discount applied to the list price. In this case, a 10% discount is offered by the car dealer. Therefore, the function \(f(x)\) is given as: \(f(x) = x - 0.10x = 0.90x\)

Step 2: Define the rebate function

The rebate function is defined as the list price minus the rebate amount. In this case, the manufacturer offers a $2000 rebate. Therefore, the function \(g(x)\) is given as: \(g(x) = x - 2000\)

Step 3: Apply composite function for discount before rebate

A composite function is a combination of two functions where the output of one function becomes the input for the second function. In this scenario, we are applying the discount first (using \(f(x)\)) then the rebate (using \(g(x)\)). Thus, using a list price of $18,000, the composite function is: \(g(f(18000)) = g(0.90 * 18000) = g(16200) = 16200 - 2000 = 14200\)

Step 4: Apply composite function for rebate before discount

In this scenario, we are applying the rebate first (using \(g(x)\)) then the discount (using \(f(x)\)). Thus, using a list price of $18,000, the composite functions is: \(f(g(18000)) = f(18000 - 2000) = f(16000) = 0.90 * 16000 = 14400\)

Key Concepts

Composite Functions

A composite function is like a mathematical recipe where you use the output of one function as the input for another. Imagine you have two events to attend: first a concert, then a dinner. You need to go to the concert, end it, and then head to the dinner. Similar to that idea, composite functions combine two separate processes.
To form a composite function, you have two functions, let's call them \( f(x) \) and \( g(x) \). If you apply \( f(x) \) first and then \( g(x) \), the result is written as \( g(f(x)) \). The output of \( f(x) \) is the input for \( g(x) \).
This concept is often used in situations like our car pricing example, where you want to apply both a discount and a rebate. You first calculate one, then use that result in the other calculation. In the exercise, applying the discount before the rebate involves using \( f(x) \) followed by \( g(x) \). Conversely, applying the rebate before the discount is \( f(g(x)) \).
Using composite functions helps to understand how changing the order of operations affects the final result. Depending on the order, you might end up with a different price for the car by around a few hundred dollars.

Discount and Rebate

In our car pricing scenario, two types of price reductions are applied: a discount and a rebate. Understanding their roles is crucial for calculating the actual cost of an item, especially something as pricey as a car.

• Discount: This is a percentage reduction off the list price offered directly by the dealer. In this case, the car dealer offers a 10\(\%\) discount. The formula to calculate the discount is \( f(x) = 0.90x \). Here, \(x\) represents the original price of the car.
• Rebate: Unlike a discount, a rebate is usually a fixed amount subtracted from the item's price, often provided by the manufacturer. Here, the rebate is \$2000, and the rebate function is \( g(x) = x - 2000 \).

Each component affects the final price differently. The sequence of applying these functions (first discount, then rebate, or vice versa) can also change the final cost. Thus, knowing and applying these functions correctly is essential for accurate financial calculations.

Function Notation

Function notation is a way to represent functions that makes it clear which operation is being performed on which variable. It uses the format \( f(x) \), where \( f \) is the name of the function, and \( x \) is the input variable.

• This notation helps in understanding what input goes into the function and what kind of output is expected. For instance, \( f(x) = 0.90x \) tells us that this function takes \(x\), multiplies it by 0.90, and gives the result as output. This represents our 10\(\%\) discount scenario.
• In our example, the functions \( f(x) \) and \( g(x) \) are set to run sequentially over the price of a car. First, the output of \( f(18000) = 16200 \) becomes the new input for \( g(x) \), thus simplifying complex operations.

Using function notation helps students break down complex problems into simpler, more manageable parts. It ensures mathematical expressions are clear and concise, avoiding confusion and enhancing problem-solving skills.