Chapter 1 A Beginning Library of Elementary Functions Section 1 Functions

Definition of a function A Function is a rule (process or method) that produces a correspondence between two sets of elements such that to each element in the first set there corresponds one and only one element in the second set. The first set is called the domain (x values) and the second set is called the range (y values).

Examples Function Function Not a Function 0 0 1 1 -1 2 4 -2 3 9 -3 0 0 1 1 -1 2 4 -2 3 9 -3 Domain Range -2 -8 -1 -1 0 0 1 1 2 8 Domain Range -2 4 -1 1 0 0 1 2 Domain Range Function Function Not a Function

Vertical Line Test An equation defines a function if each vertical line in the coordinate system passes through at most one point on the graph of the equation. If any vertical line passes through two or more points on the graph of an equation, then the equation does not define a function.

Examples Not a Function Not a Function Function

Functions Defined by Equations If in an equation in two variables, we get exactly one output (value for the dependent variable) for each input (value for the independent variable), then the equation defines a function. If we get more than one output for a given input, the equation does not define a function.

Examples Example 2 Example 1 Function Not a Function

Function Notation For any element x in the domain of the function f, the symbol f(x) represents the element in the range of f corresponding to x in the domain of f. If x is an input value, then f(x) is the corresponding output value. If x is an element that is not in the domain of f, then f is not defined at x and f(x) does not exist.

Function Evaluation Example 1 Example 2 Example 3 Not a Real Number

Domains and Ranges of a Function If a function is specified by an equation and the domain is not indicated, then we assume that the domain is the set of all real number replacements of the independent variable (inputs) that produce real values for the dependent variable (outputs). The range is the set of all outputs corresponding to input values.

Finding the Domain of a Function Problems Zero in the denominator Negative numbers under square roots Take values of x that cause problems out of the domain