Problem 65 - Algebra and Trigonometry

Question

Multiple Discounts An appliance dealer advertises a 10\(\%\) discount on all his washing machines. In addition, the manufacturer offers a \(\$ 100\) rebate on the purchase of a washing machine. Let \(x\) represent the sticker price of the washing machine. (a) Suppose only the 10\(\%\) discount applies. Find a function \(f\) that models the purchase price of the washer as a function of the sticker price \(x\) (b) Suppose only the \(\$ 100\) rebate applies. Find a function \(g\) that models the purchase price of the washer as a function of the sticker price \(x\) (c) Find \(f \circ g\) and \(g \circ f .\) What do these functions represent? Which is the better deal?

Step-by-Step Solution

Answer

The better deal is to apply the 10% discount first, then the $100 rebate.

Step 1: Applying the 10% Discount

To find the function that represents the purchase price with a 10\(\%\) discount, we need to multiply the sticker price \(x\) by 0.9. This represents a reduction of 10\(\%\) from the original price. So, the function \(f(x)\) is formulated as: \[ f(x) = 0.9x \] This function calculates the final price after applying a 10\(\%\) discount to the sticker price.

Step 2: Applying the $100 Rebate

When only the \(\\(100\) rebate is applied, we need to subtract \(\\)100\) from the sticker price \(x\). Thus, the function \(g(x)\) is given by: \[ g(x) = x - 100 \] This function represents the final purchase price after the rebate of \(\$100\) is subtracted from the sticker price.

Step 3: Composing Functions f and g

To evaluate \(f \circ g(x)\), substitute \(g(x)\) into \(f(x)\):\[ f(g(x)) = f(x - 100) = 0.9(x - 100) = 0.9x - 90 \] This expresses the purchase price after first applying the \(\$100\) rebate and then the 10\(\%\) discount.

Step 4: Composing Functions g and f

To find \(g \circ f(x)\), substitute \(f(x)\) into \(g(x)\):\[ g(f(x)) = g(0.9x) = 0.9x - 100 \] This represents the purchase price after first applying the 10\(\%\) discount and then subtracting the \(\$100\) rebate.

Step 5: Comparing Deals

We have two compositions: - \(f \circ g(x) = 0.9x - 90\)- \(g \circ f(x) = 0.9x - 100\)Since \(0.9x - 100 < 0.9x - 90\), \(g \circ f(x)\) results in a lower purchase price, making it the better deal.

Key Concepts

Discount Calculation

The concept of discount calculation is crucial in determining the final price a customer pays after adjusting for reductions given on the original price. In this exercise, we're looking at two types of discounts: a percentage discount and a fixed monetary rebate.

The percentage discount is expressed as a certain percent off the original price. For instance, a 10\(\%\) discount on a price of \(x\) means you multiply \(x\) by 0.9 to find the reduced price. This calculation comes from the idea that a 10\(\%\) reduction means you pay 90\(\%\) of the original price. Thus, the function for a 10\(\%\) discount can be represented as:

• Multiply the original price by 0.9: \(f(x) = 0.9x\)

On the other hand, a rebate is a fixed amount subtracted from the original price. Here, a \(\\(100\) rebate implies that \(\\)100\) is simply deducted from the sticker price \(x\). The resulting function is:

• Subtract \(\$100\) from the sticker price: \(g(x) = x - 100\)

Understanding how to apply each type of discount helps in calculating the final selling price effectively.

Function Composition

Function composition involves combining two or more functions to form a new function. This is done by using the output of one function as the input of another. In our exercise, function composition is used to combine the effects of both the discount and the rebate to see how they work together.

Let's consider using the functions for the discount and rebate. First, if we apply the \(\\(100\) rebate and then the 10\(\%\) discount, the function composition becomes:

• \(f(g(x)) = f(x - 100) = 0.9(x - 100) = 0.9x - 90\)

This means the purchase price after applying the rebate first and discount second is calculated this way.

Conversely, if we apply the 10\(\%\) discount first followed by the \(\\)100\) rebate, the composition is:

• \(g(f(x)) = g(0.9x) = 0.9x - 100\)

By composing functions, we can explore different ways discounts and rebates interact, giving us insights into pricing strategies.

Mathematical Modeling

Mathematical modeling is the process of representing real-world scenarios using mathematical concepts and language. It allows us to analyze situations systematically and make informed decisions. In the context of discount calculations, mathematical modeling helps us understand how different discount strategies affect pricing.

By setting up functions for each type of discount (percentage and rebate), we create a mathematical model that can be applied across various scenarios to calculate final prices:

• The percentage discount model: \(f(x) = 0.9x\)
• The rebate model: \(g(x) = x - 100\)

These models are combined further through function composition, providing all possible permutations of discounts being applied.

Ultimately, mathematical modeling in this context aids in judging which pricing strategy offers the bigger saving to customers, allowing businesses to make strategic decisions on discount offerings.