Composite Function

Algebra-2

1. Fundamental Concepts

Definition: A composite function is a function that uses the output of one function as the input of another function.
Notation: If \( f \) and \( g \) are functions, then the composite function \( (f \circ g)(x) = f(g(x)) \).
Order of Operations: The order in which functions are composed matters; generally, \( (f \circ g)(x) \neq (g \circ f)(x) \).

2. Key Concepts

Basic Rule: $$(f \circ g)(x) = f(g(x))$$

Degree Preservation: The domain of \( (f \circ g)(x) \) is all \( x \) in the domain of \( g \) such that \( g(x) \) is in the domain of \( f \).

Application: Used to model real-world scenarios where one process depends on the outcome of another.

3. Examples

Example 1 (Basic)

Problem: Given \( f(x) = 2x + 3 \) and \( g(x) = x^2 \), find \( (f \circ g)(x) \).

Step-by-Step Solution:

Substitute \( g(x) \) into \( f(x) \): \( (f \circ g)(x) = f(g(x)) = f(x^2) \).
Apply \( f \) to \( x^2 \): \( f(x^2) = 2(x^2) + 3 = 2x^2 + 3 \).

Validation: Substitute \( x = 1 \) → Original: \( f(1^2) = 2(1) + 3 = 5 \); Simplified: \( 2(1)^2 + 3 = 5 \) ✓

Example 2 (Intermediate)

Problem: Given \( f(x) = \sqrt{x} \) and \( g(x) = x - 4 \), find \( (f \circ g)(x) \).

Step-by-Step Solution:

Substitute \( g(x) \) into \( f(x) \): \( (f \circ g)(x) = f(g(x)) = f(x - 4) \).
Apply \( f \) to \( x - 4 \): \( f(x - 4) = \sqrt{x - 4} \).

Validation: Substitute \( x = 8 \) → Original: \( f(8 - 4) = \sqrt{4} = 2 \); Simplified: \( \sqrt{8 - 4} = 2 \) ✓

4. Problem-Solving Techniques

Visual Strategy: Use function diagrams to visualize the composition process.
Error-Proofing: Always check the domains of the functions involved to ensure they are compatible.
Concept Reinforcement: Practice with various types of functions (linear, quadratic, square root, etc.) to reinforce understanding.