Relations and Functions

Definition and Key Concepts

In mathematics, a relation is a set of ordered pairs, while a function is a special type of relation in which each input (independent variable) is associated with exactly one output (dependent variable). Understanding the distinction between relations and functions is fundamental in precalculus.

• Dependent variable: The variable whose value depends on another variable.

• Independent variable: The variable that determines the value of the dependent variable.

• Function: A relation in which each input has exactly one output.

• Input-output (function) machine: A conceptual model for functions, where each input produces a unique output.

Example: The amount paid for gas at a station depends on the number of gallons purchased (input: gallons, output: cost).

Relation vs. Function Table

Function Notation

Functions are often written using function notation: , which is read as "f of x" or "the value of f at x." The symbol is just another name for the dependent variable .

• Example: If , then .

Domain and Range

Definitions

The domain of a relation or function is the set of all possible input values (independent variable), and the range is the set of all possible output values (dependent variable).

• Domain: All values of for which the function is defined.

• Range: All values of that the function can produce.

Example: For the relation , the domain is and the range is .

Agreement on Domain

Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real outputs when substituted for the independent variable.

Determining Whether Relations Are Functions

Vertical Line Test

The vertical line test is a graphical method to determine if a relation is a function. If every vertical line intersects the graph at no more than one point, the relation is a function.

• Example: The graph of passes the vertical line test and is a function. A circle fails the test and is not a function.

Identifying Functions, Domains, and Ranges

Examples

• Linear function: ; domain: , range:

• Quadratic function: ; domain: , range:

• Absolute value function: ; not a function of

• Rational function: ; domain:

Increasing, Decreasing, and Constant Functions

Definitions

A function defined on an interval can be classified as increasing, decreasing, or constant:

• Increasing: increases on if, whenever , .

• Decreasing: decreases on if, whenever , .

• Constant: is constant on if, for every and , .

Example: On a graph of the height of a kite over time, intervals where the graph rises indicate increasing height, intervals where it falls indicate decreasing height, and flat intervals indicate constant height.

Basic Concepts of Linear Functions

Definition of Linear Function

A linear function is a function of the form , where and are real numbers. If , the domain and range are both .

• Example:

Standard Form of a Linear Equation

The standard form of a linear equation is , where , , and are real numbers and and are not both zero.

Graphing Linear Functions

• Intercepts: The points where the graph crosses the axes.

• Vertical line: ; domain: , range:

• Horizontal line: ; domain: , range:

Slope

Definition and Formula

The slope of a line measures its steepness and is defined as the ratio of the change in to the change in between two points and :

• Slope formula: , where

Example: For points and ,

Slopes of Horizontal and Vertical Lines

• Horizontal line: Slope is $0$

• Vertical line: Slope is undefined

Properties of Slope

• A line with positive slope rises from left to right.

• A line with negative slope falls from left to right.

• A line with zero slope is horizontal.

• A line with undefined slope is vertical.

Average Rate of Change

Definition

The average rate of change of a function over the interval is given by:

This represents the slope of the secant line joining the points and on the graph of .

Summary Table: Types of Functions and Their Properties

Applications and Examples

• Modeling: Functions are used to model real-world relationships, such as cost, profit, and physical phenomena.

• Graph interpretation: Understanding the shape and behavior of graphs helps in analyzing functions and their properties.

Additional info: Some context and definitions have been expanded for clarity and completeness, including the summary tables and explicit examples for each concept.