Problem 91 - Intermediate Algebra

Question

Exercise 89 illustrates a chain of dependence between the cost of the gasoline, the gasoline consumed, and the miles driven. Describe another chain of dependence that could be represented by a composition function.

Step-by-Step Solution

Answer

A composition function can be established from hours worked to performance, then linked to earnings.

Step 1: Identify Key Elements

Begin by identifying three components that are dependent on each other. For example: hours worked, hourly wage, and total earnings.

Step 2: Establish Initial Dependency

The total earnings depend on the hours worked and the hourly wage. This can be represented by the function: \( E(h, w) = h \times w \), where \( E \) is the total earnings, \( h \) is the hours worked, and \( w \) is the hourly wage.

Step 3: Introduce Additional Dependency

Consider the dependency between job performance and hours worked: \( P(h) = f(h) \), where \( P \) represents job performance as a function of hours worked. This shows that job performance depends on how many hours are worked.

Step 4: Composite Function Formation

The composite function now links job performance to total earnings through hours worked. This can be described as \( E(P(h), w) = P(h) \times w \), representing how total earnings depend on performance, which in turn depends on hours worked.

Step 5: Verify Logical Consistency

Ensure the chain logically connects: as hours affect performance, which impacts earnings. This checks if the relationships between each element are clear and make sense.

Key Concepts

Dependency Chains

Dependency chains are like sequences that show how different elements are connected through dependencies. Think of them as a domino effect where one element influences the next. Just like in Exercise 89 with gasoline, in our example of a workplace scenario, hours worked, and their resulting performance, and earnings can create a dependency chain.

• Start with the hours worked. The number of hours you put in directly affects job performance. More hours could mean better performance, up to a point.
• The next link is when this job performance affects your earnings. If you perform better, you might earn more, maybe through bonuses or efficiency rewards.
• Finally, the hours worked also have a direct effect on earnings since total earnings are the product of hours and hourly wage.

By understanding these chains, students can see how changing one item impacts others. It's about seeing the whole picture, rather than just isolated parts.

Function Relationships

In mathematics, functions explain how one thing changes with respect to another. When you have a function relationship, you often see how inputs (like hours worked) affect outputs (like earnings). A composition function is when you combine two functions, using the output of one as the input for another.

• In our scenario, the primary relationship is between hours and earnings. Hours worked affects total earnings, defined by the equation: \( E(h, w) = h \times w \). Here, \( h \) is the input, affecting the output \( E \).
• There's also a secondary function relationship where performance \( P \) depends on hours \( h \). This is \( P(h) = f(h) \).
• By linking \( P(h) \) into the first function, you create a composite function: \( E(P(h), w) = P(h) \times w \). You’re seeing function relationships stack together like building blocks.

It’s all about how various inputs affect outputs within a system, painting a complex picture of interactions.

Mathematical Modeling

Mathematical modeling is like crafting a blueprint of real-world scenarios using math. In our example, mathematical modeling connects hours, performance, and earnings into one coherent system. It transforms real-world scenarios into mathematical equations that help us make predictions and conclusions.

• This modeling starts by identifying the real-world elements you want to connect: here, it's the working hours, performance, and pay.
• Next, you create functions to express these relationships mathematically. These functions show how changes in one factor affect others, creating a predictive model.
• Finally, you check the logical consistency of your model. Does the math make sense with what you know about the real world?

Modeling in math is about simplifying complexities into understandable chunks. It allows you to simulate and anticipate outcomes based on different inputs.