This section explores the concept of function composition and its notation methods.

Definition: Function composition is the application of one function to the results of another, written as (f∘g)(x) or f(g(x)).

Vocabulary: "f∘g" is read as "f of g" or "g into f"

Example: For f(x)=3x+2 and g(x)=2x-1, the composition (f∘g)(x)=6x-1

The page emphasizes:

• Different notation methods for composition
• Step-by-step process of composing functions
• Importance of understanding input and output relationships

This page details the process of evaluating function composition with specific values.

Highlight: To find (f∘g)(value), first calculate g(value), then use that result as input for f.

Example: For f(x)=x²+4 and g(x)=2x, to find (f∘g)(2):

1. Calculate g(2)=4
2. Then calculate f(4)=20

The page covers:

• Composition with numerical values
• Composition with variables
• Multiple examples of both types

This section explores important properties and characteristics of function composition.

Quote: "In most cases (f∘g)(x)≠(g∘f)(x) therefore composition of functions is not commutative."

Example: Using ordered pairs and graphs to demonstrate composition: For f={(2,3),(-1,1),(0,0)} and g={(-3,1),(-1,-2),(0,2)}, find (f∘g)(0)

The page includes:

• Visual representations of function composition
• Working with ordered pairs
• Non-commutative property demonstration

This page presents more complex operations and compositions with various function types.

Highlight: When finding composite functions, always state necessary restrictions in the domain.

Example: For f(x)=2x+5 and g(x)=x²-1: (f∘g)(x)=2$x²-1$+5=2x²-2+5=2x²+3

The content covers:

• Multiple function operations
• Complex compositions
• Domain restrictions
• Practical applications

The final page provides additional practice problems and real-world applications.

Example: Given f(x)=4x-1, j(x)=x²-6x, k(x)=-x+4: $f+j$(x)=x²-2x-1

Highlight: Pay special attention to restrictions when working with composite functions.

The page includes:

• Comprehensive practice problems
• Multiple-step compositions
• Domain restriction analysis
• Complex function operations

This page introduces fundamental operations with functions, focusing on sum, product, difference, and quotient operations.

Definition: Operations with functions involve combining two functions using basic mathematical operations to create new functions.

Example: For functions f(x)=4x²+6x-9 and g(x)=6x²-x+2, the sum $f+g$(x) = 10x²+5x-7

Highlight: When performing division of functions, always remember to state the restriction g(x)≠0.

The page demonstrates several key operations:

• Addition and subtraction of polynomial functions
• Multiplication of functions
• Division with attention to restrictions