Functions and Their Properties

Definition of a Function

A function is a correspondence between a first set, called the domain, and a second set, called the range, such that each member of the domain corresponds to exactly one member of the range.

• Domain: The set of all possible input values (independent variable).

• Range: The set of all possible output values (dependent variable).

• Function notation: If y is a function of x, we write .

Finding Function Values

Functions can be described by equations, such as or . To graph the function given by , we find ordered pairs by substituting values for x:

• For : → The graph includes (4, 11).

• For : → The graph includes (-2, -1).

• For : → The graph includes (5, 13).

The input members of the domain are the values of x substituted into the equation. The output members (of the range) are the resulting values of y for each value of x.

Function notation is often used: means "the value of the function f at x is ".

Example: Evaluating a Function

• Given , find , , and :

Graphs of Functions

Definition of the Graph of a Function

The graph of a function is a drawing that represents all the input-output pairs . When the function is given by an equation, the graph of the function is the graph of the equation .

• Each point on the graph corresponds to an input x and its output .

• Example: For , the point (3, 9) is on the graph because .

The Vertical-Line Test

The Vertical-Line Test is a graphical method to determine if a curve is the graph of a function. A graph represents a function if and only if it is impossible to draw a vertical line that intersects the graph more than once.

• If any vertical line crosses the graph more than once, the graph does not represent a function.

• Example: The graph of passes the vertical-line test, but the graph of a circle does not.

Piecewise-Defined Functions

Some functions are defined by different expressions depending on the value of the input variable. These are called piecewise-defined functions.

• Example:

• For , ; for , .

Section Summary

• A function is a correspondence between two sets such that for each member of the first set (the domain), there corresponds exactly one member of the second set (the range).

• A function's domain represents inputs, and its range represents outputs.

• A function given as an equation can be written using function notation: , where is the name of the function. Ordered pairs are of the form .

Additional info:

• The notes use standard terminology and examples found in introductory calculus and precalculus courses.

• Piecewise functions and the vertical-line test are foundational for understanding more advanced calculus topics such as limits and continuity.