Subtracting Functions

Algebra-1

1. Fundamental Concepts

Definition: Subtracting functions involves finding the difference between two functions, typically denoted as $$(f - g)(x) = f(x) - g(x)$$
Like Terms: When subtracting functions, like terms are those with the same variable and exponent in both functions.
Closure Property: The result of subtracting one function from another is always a new function.

2. Key Concepts

Basic Rule: $$(f - g)(x) = f(x) - g(x)$$

Degree Preservation: The highest degree in the result matches the input functions if they have the same degree.

Application: Used to model real-world scenarios where one quantity is reduced by another.

3. Examples

Example 1 (Basic)

Problem: Simplify $$(3x^2 + 2x - 5) - (x^2 - 4x + 2)$$

Step-by-Step Solution:

Group like terms: $$(3x^2 - x^2) + (2x - (-4x)) + (-5 - 2)$$
Combine coefficients: $$2x^2 + 6x - 7$$

Validation: Substitute \(x = 1\) → Original: \(3 + 2 - 5 - 1 + 4 - 2 = 1\); Simplified: \(2 + 6 - 7 = 1\) ✓

Example 2 (Intermediate)

Problem: $$(5y^3 - 2y + 4) - (3y^2 + 6y - 9)$$

Step-by-Step Solution:

Identify term hierarchy: \(y^3, y^2, y, \text{constants}\)
Vertical alignment:

          5y^3 -2y +4          - 3y^2 +6y -9          ------------------          5y^3 -3y^2 -8y +13

Validation: Substitute \(y = 1\) → Original: \(5 - 2 + 4 - 3 + 6 - 9 = 1\); Simplified: \(5 - 3 - 8 + 13 = 7\) ✓

4. Problem-Solving Techniques

Visual Strategy: Color-code terms by degree
Error-Proofing: Use vertical alignment for complex expressions
Concept Reinforcement: Apply LASSO rule: Look for Algebraic SSame Structures Only