Problem 60 - Algebra and Trigonometry

Question

Multiple Discounts You have a \(\$ 50\) coupon from the manufacturer good for the purchase of a cell phone. The store where you are purchasing your cell phone is offering a 20\(\%\) discount on all cell phones. Let \(x\) represent the regular price of the cell phone. (a) Suppose only the 20\(\%\) discount applies. Find a function \(f\) that models the purchase price of the cell phone as a function of the regular price \(x .\) (b) Suppose only the \(\$ 50\) coupon applies. Find a function \(g\) that models the purchase price of the cell phone as a function of the sticker price \(x .\) (c) If you can use the coupon and the discount, then the purchase price is either \(f \circ g(x)\) or \(g\) o \(f(x),\) depending on the order in which they are applied to the price. Find both \(f \circ g(x)\) and \(g \circ f(x) .\) Which composition gives the lower price?

Step-by-Step Solution

Answer

Apply the discount first and then the coupon for the lower price.

Step 1: Define the discount function

The store offers a 20% discount on the regular price of the cell phone. The price after the discount can be calculated as 80% of the regular price. Therefore, the function that models the purchase price after the discount is given by \( f(x) = 0.8x \).

Step 2: Define the coupon function

The manufacturer provides a \(50 coupon. The function that models the purchase price after applying the coupon is found by subtracting \)50 from the regular price, so \( g(x) = x - 50 \).

Step 3: Find the purchase price applying the coupon first

First, apply the coupon to the regular price, resulting in the price \( g(x) \). Then, apply the 20% discount to this discounted price. Thus, the combined function \( f \circ g(x) \) is given by \( f(g(x)) = f(x - 50) = 0.8(x - 50) = 0.8x - 40 \).

Step 4: Find the purchase price applying the discount first

First, apply the 20% discount to the regular price, which gives \( f(x) \). Then subtract $50 from this discounted price using the coupon. Thus, the combined function \( g \circ f(x) \) is given by \( g(f(x)) = g(0.8x) = 0.8x - 50 \).

Step 5: Compare the results

Now, compare \( f \circ g(x) = 0.8x - 40 \) and \( g \circ f(x) = 0.8x - 50 \). Since \( 0.8x - 50 \) is less than \( 0.8x - 40 \), applying the discount first and then the coupon gives the lower purchase price.

Key Concepts

Discount Function

In the world of shopping, discounts provide great opportunities for savings. When it comes to function composition, understanding how to model these discounts mathematically can make these savings clear and predictable.

Imagine a 20% discount offered by a store. This can be translated into a mathematical function. Here's how: if the original price of a product is represented by the variable \( x \), applying a 20% discount would mean calculating what 80% of that price is. Therefore, the function \( f(x) = 0.8x \) models the discounted price. This function reduces the original price by multiplying it with 0.8, effectively applying the 20% discount.

Key points to remember:

• The discount function scales the original price to reflect the percentage saved.
• In this case, each dollar from the original price is multiplied by 0.8 to determine the new price.

This approach can be adapted to other percentage-based discounts simply by changing the multiplier.

Coupon Function

Coupons often offer a fixed reduction in price, making them another useful tool in cost-saving strategies. When translated into function form, they are simple and straightforward.

Let's use a \( \(50 \) coupon on any purchase that exceeds this amount. To represent this, we have the function \( g(x) = x - 50 \). Here, \( x \) symbolizes the original price of the item, and the coupon subtracts 50 from this total. This function emphasizes that after applying the coupon, the effective purchase price decreases by \( \)50 \), assuming \( x \) is at least \( $50 \).

Here’s what’s essential about coupon functions:

• They subtract a fixed number (the coupon's value) from the original price.
• Unlike percentage discounts, coupons do not scale with the initial purchase amount; they provide the same discount irrespective of the original price.

This makes coupon functions super handy for quick calculations.

Purchase Price Calculation

When both a discount and coupon are at play, the order of application significantly influences the final price. Function composition is a method used to determine this order's impact.

In our scenario, we have a 20% discount and a \( $50 \) coupon. You may apply the coupon before the discount, or vice versa. Here's what happens with each:

• Applying the coupon first: Use \( g(x) \) and then \( f(g(x)) \). The result is \( f(g(x)) = 0.8(x - 50) = 0.8x - 40 \).
• First applying the discount: Use \( f(x) \) then \( g(f(x)) \). This results in \( g(f(x)) = 0.8x - 50 \).

By comparing \( 0.8x - 40 \) and \( 0.8x - 50 \), it becomes evident that \( 0.8x - 50 \) yields a lower price than \( 0.8x - 40 \). Therefore, applying the discount first leads to more savings.

The composition of functions illustrates these pricing interactions effectively, revealing the real impact of different savings combinations.