Question

Discounts The manufacturer of a computer is offering two discounts on last year's model computer. The first discount is a \(\$ 200\) rebate and the second discount is \(20 \%\) off the regular price, \(p\) (a) Write a function \(f\) that represents the sale price if only the rebate applies.(b) Write a function \(g\) that represents the sale price if only the \(20 \%\) discount applies. (c) Find \(f \circ g\) and \(g \circ f\). What does each of these functions represent? Which combination of discounts represents a better deal for the consumer? Why?

Short answer

The better deal is \(f(g(p)) = 0.8p - 200\) because it results in a lower price.

Step-by-step solution

Write the rebate function

When only the rebate of \(\text{\$ 200}\) is applied, the function representing the sale price is \(f(p) = p - 200\). This function subtracts \(\text{\$ 200}\) from the regular price, \(p\).

Write the percentage discount function

The function representing the sale price when only the \(20\%\) discount is applied is \(g(p) = 0.8p\). This function multiplies the regular price, \(p\), by \(0.8\) to give a sale price after a \(20\%\) discount.

Find \(f \circ g\)

\(f \circ g\) means applying function \(g\) first and then applying function \(f\) to the result. \ Start with \(g(p)\): \ \(g(p) = 0.8p\) \ Now apply \(f\) to \(g(p)\): \ \(f(g(p)) = f(0.8p) = 0.8p - 200\).

Find \(g \circ f\)

\(g \circ f\) means applying function \(f\) first and then applying function \(g\) to the result. \ Start with \(f(p)\): \ \(f(p) = p - 200\) \ Now apply \(g\) to \(f(p)\): \ \(g(f(p)) = g(p - 200) = 0.8(p - 200) = 0.8p - 160\).

Compare the functions \(f \circ g\) and \(g \circ f\)

To determine which combination of discounts is better, compare \(f(g(p))\) and \(g(f(p))\). \ \(f(g(p)) = 0.8p - 200\) \ \(g(f(p)) = 0.8p - 160\) \ Since \(0.8p - 200 < 0.8p - 160\), \(0.8p - 200\) (or \(f(g(p))\)) represents a lower price, which is a better deal for the consumer.

Key concepts

Function Composition

Function composition involves applying one function and then applying another function to the result.
It is represented by \(f \circ g\), where you apply function g first and then function f. In the context of discount calculations, function composition allows us to find the final price when multiple discounts are applied sequentially.
For instance, if you start with function \(g(p)\) which applies a 20% discount to the original price \(p\), and then apply function \(f(p)\) which subtracts a \$200 rebate, you get the combined effect in the form \(f(g(p))\).
Let's break down the steps for better understanding:

• First, apply the percentage discount function: \(g(p) = 0.8p\)
• Next, apply the rebate function to the result: \(f(0.8p) = 0.8p - 200\)

This sequence leads us to the composite function \(f \circ g(p) = 0.8p - 200\). Similarly, you can reverse the order by first applying the rebate function and then the percentage discount, leading to \(g \circ f(p) = 0.8(p - 200) = 0.8p - 160\). By comparing these results, we can determine which combination of discounts provides a lower final price, hence a better deal.

Discount Calculation

Calculating discounts accurately is crucial when dealing with price reductions.
In this exercise, two types of discounts are applied to a computer's regular price \(p\).
First, let's understand the \(200 rebate. A rebate is a fixed discount subtracted directly from the price. If the regular price is \(p\), the discount function \(f(p) = p - 200\) represents this rebate.
So, if the original price is \)1000, applying the rebate gives a new price of \(1000 - 200 = 800\).
Secondly, a percentage discount represents a proportional reduction. A 20% discount means you pay 80% of the original price. The function \(g(p) = 0.8p\) reflects this.
For instance, if the original price is \$1000, a 20% discount reduces it to \(0.8 \times 1000 = 800\).
Accurate discount calculations help determine the best price and influence purchasing decisions effectively.

Rebate and Percentage Discount

Both rebates and percentage discounts are common ways to reduce prices, and they work differently.
A rebate is a fixed amount subtracted directly from the price. For example, in function \(f(p) = p - 200\), \$200 is taken off regardless of the price.
In contrast, a percentage discount reduces the price proportionally. In function \(g(p) = 0.8p\), 20% off the price means that if the price doubles, the discount also doubles. This makes percentage discounts more flexible relative to the price.
In our exercise, combining these discounts sequentially gives us different final prices depending on the order:

• Rebate first: \(g \circ f(p) = 0.8(p - 200) = 0.8p - 160\)
• Percentage discount first: \(f \circ g(p) = 0.8p - 200\)

You can see that applying the percentage discount first followed by the rebate results in a lower final price, making it a better deal for the consumer.