Adding Functions - Algebra-1

1. Fundamental Concepts

Definition: Functions can be combined by adding their outputs for the same input value.
Like Terms: When adding functions, like terms refer to the functions being added together with the same input variable.
Closure Property: The result of adding two functions is always another function.

2. Key Concepts

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

Degree Preservation: The degree of the resulting function does not necessarily match the degrees of the original functions but depends on their combination.

Application: Used in various applications such as modeling real-world phenomena and solving complex equations.

3. Examples

Example 1 (Basic)

Problem: Simplify $$(f + g)(x) \text{ where } f(x) = 3x^2 + 2x \text{ and } g(x) = x^2 - 4x$$

Step-by-Step Solution:

Group like terms: $$3x^2 + x^2 + 2x - 4x$$
Combine coefficients: $$4x^2 - 2x$$

Validation: Substitute $x=1$ → Original: $3+2+1-4=2$ ; Simplified: $4-2=2$ ✓

Example 2 (Intermediate)

Problem: $$(f + g)(x) \text{ where } f(x) = 5y^3 - 2y + 4 \text{ and } g(x) = 3y^2 + 6y - 9$$

Step-by-Step Solution:

Identify term hierarchy: $y^3$ , $y^2$ , $y$ , constants
Vertical alignment:

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

Validation: Substitute $y=1$ → Original: $5-2+4+3+6-9=7$ ; Simplified: $5+3+4-5=7$ ✓

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