Problem 102 - Precalculus Essentials

Question

?. The regular price of a pair of jeans is \(x\) dollars. Let \(f(x)=x-5\) and \(g(x)=0.6 x\) a. Describe what functions \(f\) and \(g\) model in terms of the price of the jeans. b. Find \((f \circ g)(x)\) and describe what this models in terms of the price of the jeans. c. Repeat part (b) for \((g \circ f)(x)\) d. Which composite function models the greater discount on the jeans, \(f \circ g\) or \(g \circ f ?\) Explain.

Step-by-Step Solution

Answer

Functions \(f\) and \(g\) model a $5 and a 40% discount on the jeans' price, respectively. Their compositions \((f \circ g)(x) = 0.6x - 5\) and \((g \circ f)(x) = 0.6x - 3\) model different orders of the same discounts. The composite function \((f \circ g)(x)\) models the greater discount.

Step 1: Understand Model Functions

The regular price of the jeans is represented by \(x\). The function \(f(x)=x-5\) models a $5 discount off the regular price, while the function \(g(x)=0.6x\) represents a 40% discount off the price of the jeans.

Step 2: Compute \((f \circ g)(x)\)

The composition \(f \circ g\) means applying function g first, then applying f to the result. It is given by \((f \circ g)(x) = f(g(x)) = f(0.6x) = 0.6x - 5\). This represents a scenario where the 40% discount is applied first and then a $5 discount is applied on the reduced price.

Step 3: Compute \((g \circ f)(x)\)

Now, for the composition \(g \circ f\), function f is applied first and then g. It is given by \((g \circ f)(x) = g(f(x)) = g(x-5) = 0.6(x-5) = 0.6x - 3\). This represents a different scenario where a $5 discount is given first followed by the 40% discount on the reduced price.

Step 4: Compare \((f \circ g)(x)\) and \((g \circ f)(x)\)

The last part of the problem asks us to determine which composite function models the greater discount. This is simply a comparison between \((f \circ g)(x)\) and \((g \circ f)(x)\). Any \(x\) would suffice to test this out. If we, for example, set \(x = 10\), then \((f \circ g)(10) = 0.6*10-5 = 1\), and \((g \circ f)(10) = 0.6*(10-5) = 3\), which means that \((f \circ g)(x)\) is the greater discount.

Key Concepts

Function Composition

Function composition involves combining two functions to form a new function. It’s like following two steps where each function represents one step in the process. For example, when we have two functions, say, \(f(x)\) and \(g(x)\), composing them like \((f \circ g)(x)\) means that we first apply \(g(x)\) and then \(f\) on the result of \(g(x)\).

• In the original exercise, \(f(x) = x - 5\) models a \(5 discount and \(g(x) = 0.6x\) models a 40% discount on a pair of jeans.
• The composition \((f \circ g)(x)\) means applying the 40% discount first, then the \)5 discount.

Understanding function compositions allows us to model real-world scenarios where multiple operations occur in sequence, such as sequential discounts like in the jeans price model.

Price Discount Models

A price discount model is used to calculate the reduced price when various discounts are applied. It's not just about subtracting a percentage or amount; the sequence in which they’re applied can make a difference. Let's break it down using the given functions:

• Function \(f(x) = x - 5\) takes off \(5 from the price, so if the jeans were \)20, they become \(15.
• Function \(g(x) = 0.6x\) means after a 40% discount, if jeans were \)20, the price would be \(12.
• For \((f \circ g)(x)\), first, the jeans price is cut down by 40%, then \)5 is subtracted.
• For \((g \circ f)(x)\), $5 is deducted initially, and then a 40% discount is applied to the new price.

In retail, such discount models help businesses and consumers understand price reductions, making them crucial in sale strategies and purchasing decisions.

Function Operations

Function operations like addition, subtraction, multiplication, and especially composition allow us to perform complex calculations in an organized manner. Here, we're focusing on function composition, but understanding all operations is essential in mathematics.
In the exercise, function operations allow us to compute two distinct compositions:

• \((f \circ g)(x)\) which calculates first a percentage drop and then a fixed discount.
• \((g \circ f)(x)\) that implements the operations in reverse: a fixed discount followed by a percentage off.

This illustrates how the order of operations changes the final outcome. For instance, practically in purchases, applying a larger discount first could yield a greater overall reduction in price.
Clear knowledge of function operations enhances our problem-solving skills, enabling us to navigate through various mathematical challenges with confidence.