Relations and Functions Overview

Definition: A relation is a set of ordered pairs (x, y), where x is an input and y is an output.
Types of Relations:
- One-to-One: Each element of the domain maps to a unique element in the range.
- Many-to-One: Multiple elements in the domain map to a single element in the range.
- Onto: Every element in the range is mapped by at least one element in the domain.
- One-to-Many: An element in the domain maps to multiple elements in the range (not a function).
Representation:
- Mapping Diagrams: Visual representations of relations using arrows.
- Tables: Listing pairs of values for x and y.
- Graphs: Plotting ordered pairs on a Cartesian plane.

Definition: A special type of relation where each input has exactly one output.
Notation: Often denoted as f(x), where f is the function name and x is the input.
Types of Functions:
- Linear Functions: Can be expressed in the form f(x) = mx + b (m = slope, b = y-intercept).
- Quadratic Functions: Can be expressed as f(x) = ax² + bx + c (a, b, c are constants).
- Polynomial Functions: Sum of terms, each consisting of a variable raised to a non-negative integer power.
- Rational Functions: Ratio of two polynomials.
- Exponential Functions: f(x) = a * b^x (a ≠ 0, b > 0).
- Logarithmic Functions: Inverse of exponential functions, f(x) = log_b(x).

Domain: The set of all possible input values (x) for the function.
Range: The set of all possible output values (y) of the function.
Continuity: A function is continuous if there are no breaks or holes in its graph.
Increasing/Decreasing: A function is increasing if f(x1) < f(x2) for x1 < x2; decreasing if f(x1) > f(x2).
Intercepts:
- x-intercept: The point where the graph crosses the x-axis (f(x) = 0).
- y-intercept: The point where the graph crosses the y-axis (x = 0).

Axes: X-axis (horizontal), Y-axis (vertical).
Plotting Points: Marking (x, y) pairs on the graph.
Shape: Determined by the function type (linear straight line, parabolic curve for quadratics, etc.).
Transformations:
- Translations: Shifting the graph horizontally or vertically.
- Reflections: Flipping the graph over a line (x-axis or y-axis).
- Stretching/Compressing: Changing the width of the graph.

Composite Functions: Combining two functions, written as (f ∘ g)(x) = f(g(x)).
Inverse Functions: A function that reverses the effect of the original function, written as f⁻¹(x); satisfies f(f⁻¹(x)) = x.

A relation consists of ordered pairs (x, y), where x is the input and y is the output.
Types of Relations:
- One-to-One: Unique mapping; every domain element corresponds to a distinct range element.
- Many-to-One: Multiple domain elements map to a single range element.
- Onto: Every range element has at least one corresponding domain element.
- One-to-Many: A domain element can relate to multiple range elements; not classified as a function.
Representation Methods:
- Mapping Diagrams: Use arrows to display relationships visually.
- Tables: List pairs of x and y values systematically.
- Graphs: Plot ordered pairs on a Cartesian plane for visual representation.

Characterized as relations where each input corresponds to exactly one output.
Notation: Typically expressed as f(x), indicating the function's name and input.
Types of Functions:
- Linear Functions: Expressed in the form f(x) = mx + b, where m represents slope and b the y-intercept.
- Quadratic Functions: Formulated as f(x) = ax² + bx + c, with a, b, c as constants.
- Polynomial Functions: Consist of a sum of terms with variables raised to non-negative integer powers.
- Rational Functions: Represent the quotient of two polynomial expressions.
- Exponential Functions: Represented as f(x) = a * b^x, where a is a non-zero constant, and b is a positive base.
- Logarithmic Functions: Functions satisfying f(x) = log_b(x), which are the inverses of exponential functions.

Domain: The complete set of potential input values (x) allowable for the function.
Range: The complete set of potential output values (y) produced by the function.
Continuity: A function is considered continuous if its graph is free of breaks or gaps.
Monotonicity: A function is classified as increasing if f(x1) < f(x2) for x1 < x2; it is decreasing if f(x1) > f(x2).
Intercepts:
- x-intercept: The point on the graph where it intersects the x-axis (where f(x) = 0).
- y-intercept: The point where the graph intersects the y-axis (where x = 0).

Axes Configuration: The horizontal line is the x-axis, while the vertical line is the y-axis.
Plotting Points: Points (x, y) are marked based on their coordinates on the graph.
Graph Shape: The graph's visual form varies according to the function type, e.g., straight lines for linear functions and parabolas for quadratics.
Transformations:
- Translations: Moving the graph up, down, left, or right without changing its shape.
- Reflections: Flipping the graph over a specified axis, either x-axis or y-axis.
- Stretching/Compressing: Altering the width or height of the graph compared to its original form.

Composite Functions: Formed by combining two functions, expressed as (f ∘ g)(x) = f(g(x)), indicating that g(x) is evaluated first, followed by f.
Inverse Functions: Functions that effectively reverse the action of another function, denoted as f⁻¹(x); satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f.