Problem 64 - Algebra and Trigonometry

Question

Multiple Discounts You have a S50 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 \(g \circ f(x),\) 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

The lowest price is from applying discount first, \( g(f(x)) = 0.8x - 50 \).

Step 1: Define Function for Percentage Discount

To model the purchase price when a 20\(\%\) discount is applied, account for keeping 80\(\%\) of the price. Thus, the function \( f(x) \) is the regular price \( x \) reduced by 20\(\%\): \[ f(x) = 0.8x \] This represents paying 80\(\%\) of the original price.

Step 2: Define Function for Coupon Discount

To model the purchase price with a \( \\(50 \) coupon, subtract \( \\)50 \) from the regular price \( x \). The function \( g(x) \) is given by: \[ g(x) = x - 50 \] This formula represents the price reduction directly by \( \$50 \).

Step 3: Apply Composition to First Method

Apply the \( \$50 \) coupon first, then a 20\(\%\) discount. Compute \( f(g(x)) \) which substitutes \( g(x) \) into \( f(x) \): 1. Apply \( g(x) = x - 50 \) to get the new price after coupon.2. Apply \( 20\% \) discount using \( f \) on the value \( g(x) \):\[ f(g(x)) = 0.8(x - 50) = 0.8x - 40 \]

Step 4: Apply Composition to Second Method

Apply the 20\(\%\) discount first, then the \( \\(50 \) coupon. Compute \( g(f(x)) \) which substitutes \( f(x) \) into \( g(x) \): 1. Apply \( f(x) = 0.8x \) to get the new price after discount.2. Subtract \( \\)50 \) as modeled by \( g \):\[ g(f(x)) = 0.8x - 50 \]

Step 5: Compare Results to Determine Lower Price

Compare the final prices from the previous steps: - \( f(g(x)) = 0.8x - 40 \) - \( g(f(x)) = 0.8x - 50 \)The lower price comes from the second method, as the final formula \( g(f(x)) \) results in subtracting an additional \( \\(10 \) because the \( \\)50 \) is taken from a smaller discounted amount.

Key Concepts

Understanding Percentage Discounts

A percentage discount is a reduction in price based on a fraction of the original price. In our scenario, the store is offering a 20% discount on the cell phone. This means you will pay 80% of the regular price. This is a typical way to encourage purchases by showing a significant saving.
To model this mathematically, we use the function \( f(x) = 0.8x \). Here, \( x \) is the original price of the cell phone. We multiply \( x \) by 0.8 to reflect the 20% discount.

• This operation reduces the price by 20%.
• The function outputs the purchase price after the discount is applied.
• Think of it as taking the price and removing a certain percentage from it.

Percentage discount can be a great opportunity to save on big purchases, provided the discount is applied correctly.

Exploring Coupon Discounts

Coupon discounts offer a fixed amount off the purchase price, regardless of its original total. In our exercise, there is a \(50 coupon for the cell phone. Coupons can be more straightforward than percentage discounts because they involve subtracting a specific amount.
The function for this discount can be expressed as \( g(x) = x - 50 \). This function takes \( x \) as the regular price of the cell phone and directly subtracts \)50.

• This approach is simple and easy to understand.
• It's a flat reduction, meaning the amount saved does not change with the original price adjustment.
• Coupon discounts are especially beneficial for lower-priced items where the fixed reduction might represent a significant percentage of the cost.

Look out for deals like these, but also consider how they interact with other promos.

Price Modeling with Function Composition

Price modeling with function composition involves applying multiple discounts in specific sequences. The goal is to find which combination yields the best price. In our scenario, both a percentage discount and a coupon can be applied, allowing for two potential compositions.
The first composition \( f(g(x)) \) assumes the \(50 coupon is applied first followed by the 20% discount. This is represented as \[ f(g(x)) = 0.8(x - 50) = 0.8x - 40 \].

• Start by reducing the price by \)50, then calculate 80% of this new price.
• This method saves indirectly on the coupon amount after other discounts are applied.

The second composition \( g(f(x)) \) starts with the 20% discount, then applies the \(50 coupon. This is \[ g(f(x)) = 0.8x - 50 \].

• Start by applying the 20% discount on the full price.
• Then subtract the \)50 from this already reduced price.
• This provides a more significant saving, in this example, because the coupon is taken from a cheaper total.

Comparing the final results from these functions, \( g(f(x)) \) gives the lower price, saving an extra $10. Such compositions are valuable when dealing with multiple offers, helping consumers optimize their savings.