1. Set & Interval Notation

Sets: Collections of elements. Operations include intersection (A ∩ B), union (A ∪ B), and difference (A \ B).

Interval Notation:

• [a, b] includes both endpoints

• (a, b) excludes both endpoints

2. Relations vs. Functions

Relations: Sets of ordered pairs (x, y)

Functions: Each x-value maps to exactly one y-value

Types of Relations:

• One-to-one

• Many-to-one (function)

• One-to-many (not a function)

• Many-to-many (not a function)

3. Function Concepts

Vertical Line Test: A relation is a function if no vertical line intersects the graph more than once

Function Notation: f(x), where x is input and f(x) is output

Domain: All possible x-values

Range: All resulting y-values

4. Domain and Range

Implied Domain: Largest set where the function is defined

Restrictions by function type:

• Square roots: x ≥ 0

• Logarithms: x > 0

• Reciprocals: x ≠ 0

5. Piecewise Functions

Defined using different expressions over different intervals

Domains must not overlap

Each piece is sketched separately on its interval

6. Inverse Functions

Reverse the original function’s process

Swap x and y, solve for y

Graph reflects over the line y = x

Domain ↔ Range

Only one-to-one functions have valid inverses

7. Composite Functions

Combination of two functions: g(f(x))

Output of f becomes input for g

Defined only if range of f ⊆ domain of g

Sometimes domain restrictions are needed to make the composition valid

8. Graphing and Analysis

Includes sketching graphs of functions, inverses, and piecewise definitions

Focuses on determining domain, range, and function validity through algebra and graphs