Problem 69 - College Algebra

Question

The regular price of a computer is \(x\) dollars. Let \(f(x)=\) \(x-400\) and \(g(x)=0.75 x\) a. Describe what the functions \(f\) and \(g\) model in terms of the price of the computer. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the computer. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the computer, \(f \circ g\) or \(g \circ f ?\) Explain.

Step-by-Step Solution

Answer

The function \(f(x) = x - 400\) models a straight discount of $400 and the function \(g(x) = 0.75x\) models a 25% discount. The composite function \(f \circ g(x) = 0.75x - 400\) represents a 25% discount followed by a $400 discount. The composite function \(g \circ f(x) = 0.75(x - 400)\) represents a $400 discount followed by a 25% discount. The method that gives the greater discount depends on the original price, with \(f \circ g(x)\) giving a greater discount for prices less than $1600 and \(g \circ f(x)\) giving a greater discount for prices greater than $1600.

Step 1: Understanding the functions

From the problem, function \(f(x) = x - 400\) represents a straight discount of $400 on the price of the computer. The function \(g(x) = 0.75x\) represents a 25% discount on the price of the computer.

Step 2: Find the composite function \(f \circ g(x)\)

The composite function \(f \circ g(x)\) means we first apply function \(g\) and then function \(f\). We find that \(f \circ g(x) = f(g(x)) = f(0.75x) = 0.75x - 400\). This represents a situation where we first apply a 25% discount on the original price of the computer, and then deduct $400 from the discounted price.

Step 3: Find the composite function \(g \circ f(x)\)

The composite function \(g \circ f(x)\) means we first apply function \(f\) and then function \(g\). We find that \(g \circ f(x) = g(f(x)) = g(x - 400) = 0.75(x - 400)\). This models a situation where we first deduct $400 from the original price, and then apply 25% off the deducted price.

Step 4: Comparing the composite functions

To compare which composite function gives the greater discount, we distinguish two cases. If \(x < 1600\), then \(0.75x - 400 > 0.75(x - 400)\) and so \(f \circ g(x)\) gives the greater discount, but if \(x > 1600\), then \(0.75x - 400 < 0.75(x - 400)\) and \(g \circ f(x)\) gives the greater discount. If \(x = 1600\), both methods give the same price.

Key Concepts

Function Composition

In mathematical terms, function composition is the process of combining two or more functions in a specific order to create a new function. This process can be represented symbolically as \( (f \circ g)(x) \) where you first apply \( g \) to \( x \) and then apply \( f \) to the result. Imagine a sequence where one function hands off its output to be the input of the next function, much like a relay race. The order of functions is crucial, as reversing them could yield a different outcome. This can be seen through the lens of a discount model where different methods of applying discounts lead to separate prices for the same item.

For a practical illustration, consider our scenario involving a computer's pricing. Function \( g(x) \) applied first gives a 25% discount, reducing the price, and then function \( f(x) \) applies a fixed discount, further lowering the price. This sequential discount model showcases the utility of function composition in financial scenarios like sales and markdowns.

Algebraic Discount Model

In retail mathematics, an algebraic discount model is used to determine the final sale price of an item after applying various discounts. By using algebraic expressions to represent these discounts, businesses can calculate the impact on pricing with precision and clarity. In our exercise, two discount models are present: a flat discount with \( f(x) = x - 400 \) and a percent discount with \( g(x) = 0.75x \) which actually represents 25% off, since 0.75 is what remains after the discount is applied.

By employing these models algebraically, we can quickly analyze how different discounting sequences affect the final price. This aids consumers and retailers alike in understanding possible savings and pricing strategies. For instance, applying a percentage discount before a flat discount often yields a different final price than reversing the order - a concept that is beautifully encapsulated in the composite functions from our exercise.

Mathematical Functions

Mathematical functions are at the heart of various equations and models. They are special relationships where each input has a single output. In simple terms, a function takes in a number (input), does something to it (process), and then gives back another number (output). The functions \( f \) and \( g \) from the problem serve as perfect examples.

Function \( f(x) = x - 400 \) subtracts $400 from its input, while function \( g(x) = 0.75x \) multiplies its input by 0.75, representing a 25% reduction. These functions don't just crunch numbers; they model real-world situations, like calculating discounts, enabling us to derive actionable insights and make informed financial decisions.

Price Discount Calculations

Calculating price discounts is an essential skill, not just in shopping, but in any financial context where price adjustments are common. The process typically involves applying percentage-based or flat reductions to the original price. In our computer pricing exercise, we explored two primary discount types with functions \( f \) and \( g \) and how their composition affects overall savings.

To calculate the discounted price with a mathematical function like \( g(x) = 0.75x \) for a percentage discount or \( f(x) = x - 400 \) for a flat rate, we input the original price into the function to receive the discounted price. When stacking discounts, sequence matters profoundly, as we saw in comparing \( f \circ g \) versus \( g \circ f \) outcomes. Understanding the interplay of these discounts is not only pertinent to savvy shopping but also crucial to comprehending more complex economic models where multiple adjustments are made to a base value.