Problem 88 - Algebra and Trigonometry Real Mathematics, Real People

Question

The suggested retail price of a new car is \(p\) dollars. The dealership advertised a factory rebate of \(\$ 2000\) and a \(9 \%\) discount. (a) Write a function \(R\) in terms of \(p\) giving the cost of the car after receiving the rebate from the factory. (b) Write a function \(S\) in terms of \(p\) giving the cost of the car after receiving the dealership discount. (c) Form the composite functions \((R \circ S)(p)\) and \((S \circ R)(p)\) and interpret each. (d) Find \((R \circ S)(24,795)\) and \((S \circ R)(24,795) .\) Which yields the lower cost for the car? Explain.

Step-by-Step Solution

Answer

Functions: \(R(p) = p-2000\), \(S(p) = 0.91p\). Composite: \((R \circ S)(p) = 0.91p - 2000\), \((S \circ R)(p) = 0.91p - 1820\). \((R \circ S)(24795) = 20563.45\) and \((S \circ R)(24795) = 20731.95\). The method \((R \circ S)\) yields the lowest cost.

Step 1: Create function R(p)

The function \(R(p)\) represents the cost of the car after the factory rebate. Since the factory is offering a $2000 rebate, the cost of the car after the rebate will be \(p-2000\). Therefore, \(R(p) = p - 2000\).

Step 2: Create function S(p)

The function \(S(p)\) represents the cost of the car after the dealership discount. This is calculated by subtracting \(9\%\) of the total cost from the total cost itself. Thus, \(S(p) = p - 0.09p = 0.91p\).

Step 3: Form the composite functions

The composite function \((R \circ S)(p)\) means the function \(R\) after \(S\), which is calculated as \(R(S(p)) = R(0.91p) = 0.91p - 2000\). The second composite function \((S \circ R)(p)\) means the function \(S\) after \(R\), which is calculated as \(S(R(p)) = S(p-2000) = 0.91(p-2000) = 0.91p - 1820\).

Step 4: Interpret the composite functions

\((R \circ S)(p)\) means that you first apply the dealership discount and then the factory rebate. On the other hand, \((S \circ R)(p)\) means that the factory rebate is applied first and then the dealership discount.

Step 5: Evaluate the composite functions for \(p = 24795\)

Now, we substitute \(p = 24795\) in \((R \circ S)(p)\) and \((S \circ R)(p)\). We find \((R \circ S)(24795) = 0.91(24795) - 2000 = 20563.45\) and \((S \circ R)(24795) = 0.91(24795 - 2000) = 20731.95\). Taking a look at both values, the lowest cost is given by \((R \circ S)(24795)\). This implies that applying the dealership discount first and then the factory rebate provides the lowest price.

Key Concepts

Function Notation

Function notation is a way to describe mathematical functions in a clear and concise manner. It's like a shorthand that helps us understand what a function does to an input value. When we see something like \( f(x) = 2x + 3 \), this tells us that \( f \) is a function that takes an input \( x \) and transforms it according to the rule \( 2x + 3 \).
In the context of our exercise, we have the functions \( R(p) \) and \( S(p) \), where \( R \) and \( S \) describe how the car's price \( p \) changes after a rebate or discount. Using function notation makes it easy to address how each operation affects the price. By simply plugging in different values for \( p \), we can immediately see the outcome based on these functions.

Discount Calculation

Discount calculation is often useful when shopping, as it helps you figure out exactly how much money you'll save. In this exercise, we calculate a \(9\%\) discount on the car's price. This is done using the function \( S(p) = 0.91p \).
This formula works because when you pay 91% of the full price, you're essentially getting a 9% discount.

• First, we calculate 9% of the price: \(0.09p\).
• Then, we subtract that amount from the original price \( p \).
• This results in: \(S(p) = p - 0.09p = 0.91p\).

Understanding these steps helps when you encounter discounts in everyday situations.

Function Composition

Function composition involves combining two functions to create a new one. Think of it as a process where the output of one function becomes the input for another.
In the problem, we have two ways to combine functions:

• \((R \circ S)(p)\) means applying \(S\) first, then \(R\). Imagine taking the price after the discount and then applying the rebate.
• \((S \circ R)(p)\) involves \(R\) first, then \(S\). Here, you take the price after rebate and apply the discount.

The order matters because it changes the final outcome. This exercise helps you see how the sequence of operations affects both mathematical solutions and financial transactions.

Mathematical Modeling

Mathematical modeling uses math to represent real-world situations, helping us make predictions or decisions. Here, we model the cost of a car using functions to represent different financial incentives.
By creating models as \(R(p) = p - 2000\) and \(S(p) = 0.91p\), we translate rebates and discounts into mathematical terms. This can be incredibly helpful in financial planning and decision-making, since:

• It lets us manipulate numbers to see potential outcomes.
• We can compare different scenarios, like applying discounts before rebates.
• It allows us to predict the cost of the car under different financial offers.

Understanding mathematical modeling empowers you to make smart financial choices.