1. Chapter 1 A Beginning Library of Elementary Functions
Section 1
Functions

2. 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).

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

4. 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.

5. Examples
Not a Function
Not a Function
Function

6. 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.

7. Examples
Example 2
Example 1
Function
Not a Function

8. 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.

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

10. 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.

11. 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