Relations and Functions

Definitions 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 output value, often denoted as y.

• Independent variable: The input value, often denoted as x.

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

• Relation: Any set of ordered pairs (x, y).

Example: The amount paid for gas at a station depends on the number of gallons purchased. Here, the number of gallons is the independent variable, and the amount paid is the dependent variable.

Determining Whether Relations Are Functions

To determine if a relation is a function, check if each input value corresponds to only one output value. This can be done by examining ordered pairs or using the vertical line test on a graph.

• Vertical Line Test: If any vertical line intersects the graph at more than one point, the relation is not a function.

Example: For the relation , each x-value is paired with only one y-value, so M is a function.

Domain and Range

Definitions

The domain of a relation or function is the set of all possible input values (x-values). The range is the set of all possible output values (y-values).

• Domain: Set of all independent variable values.

• Range: Set of all dependent variable values.

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

Agreement on Domain

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

Example: The domain of is all real numbers except .

Function Notation

Using Function Notation

Function notation expresses the output value as a function of the input value. The notation means "the value of function f at x."

• Function notation:

• Example: If , then .

Finding an Expression for

To express y as a function of x, solve the equation for y in terms of x.

• Step 1: Solve the equation for y.

• Step 2: Replace y with .

Example: Given , write .

Increasing, Decreasing, and Constant Functions

Definitions

A function can be classified as increasing, decreasing, or constant over an interval based on the behavior of its output values.

• Increasing: increases over if, whenever , .

• Decreasing: decreases over if, whenever , .

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

Example: On a graph of a kite's height 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 graph is a straight line with domain and range both .

• Standard form:

• Constant function:

Example: is a linear function.

Graphing Linear Functions

To graph a linear function, plot two points and draw a straight line through them. The slope determines the steepness and direction of the line.

• Vertical line: (not a function)

• Horizontal line: (a constant function)

Slope

Definition and Formula

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

• Slope formula: , where

Example: The slope between and is .

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 an interval is the change in the output value divided by the change in the input value.

• Formula:

Example: For , the average rate of change from to is .

Tables: Summary of Key Concepts

Additional info: Some context and definitions have been expanded for clarity and completeness, including examples and formulas for each concept.