Lesson 7: Algebraic Fractions and Functional Relations

👨🏽‍🏫 Introduction

Ever tried to add or simplify a fraction with variables like x and y in it? That’s what we call an algebraic fraction! These behave just like normal fractions — but with expressions in the numerator and denominator.

We also use functional relations to describe how one quantity depends on another, especially in formulas and mappings.

By the end of this lesson, you’ll be able to:

• Simplify algebraic fractions
• Add, subtract, multiply, and divide algebraic fractions
• Evaluate and interpret simple functions

🧠 Key Concepts and Explanations

➗ What Are Algebraic Fractions?

Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions.

Examples:

🔁 Rules for Simplifying

1. Factor both numerator and denominator
2. Cancel common factors
3. State restrictions (values that make denominator zero)

Example:

Factor numerator and denominator:

Cancel one :

Note: x ≠ 3

➕ Adding and Subtracting

Find the Lowest Common Denominator (LCD), then rewrite, simplify, and combine.

Example:

LCD is

✖️ Multiplying and Dividing

• Multiply: Multiply across numerators and denominators
• Divide: Invert the second fraction and multiply

Example:

Note: x ≠ 0, 1, -1

📉 Functional Relations

🧩 What Is a Function?

A function links an input (usually x) to exactly one output (usually y).

We write it as , which means “function of x.”

Example: If , then:

↔️ Functional Mapping

Mapping diagrams and tables are used to show inputs and outputs.

🧮 Sample Problem Walkthroughs

1. Simplify:
   Factor numerator:
   Cancel 2x:
   Note: x ≠ 0
2. Add:
   LCD =
   
   Note: x ≠ 0
3. Evaluate: If , find
   

✍🏽 Practice Exercises

🅰️ A. Simplify the following:

🅱️ B. Perform the operations:

🆎 C. Evaluate and interpret:

1. If , find
2. If , find

✅ Answers and Explanations

🅰️ A. Simplify

• 1.
  Note: x ≠ -2, 3
• 2.
  Note: x ≠ 0

🅱️ B. Operations

• 1. LCD =
  
  Note: x ≠ -2, 1
• 2.
  Note: x ≠ 0, -3

🆎 C. Evaluation

• 1.
• 2.

📌 Recap

• ✅ Algebraic fractions can be simplified using factoring and cancellation.
• ✅ Operations follow the same rules as with numbers — just with variables!
• ✅ A function assigns a unique output to every input — evaluate by substituting values into the expression.

💭 Reflection Prompt

Have you seen any real-life examples where a formula or function determines an outcome — like calculating grades, costs, or distance? How could algebraic fractions help you make better decisions?