1. Relations and Functions
Lesson 1: Properties of Functions

2. Todays Objectives
Explain, using examples, why some relations are not functions but all functions are relations
Determine if a set of ordered pairs represents a function
Generalize and explain rules for determining whether graphs and sets of ordered pairs represent functions

3. Relations and Functions
A relation associates the elements of one set of data with the elements of another set of data
A function is a specific type of relation where each element in the domain is associated with exactly one element in the range. On a graph, this means that each x-value has exactly one corresponding y-value. Look at the graph on the next page for example:

4. Relation vs. Function

5. Vertical Line Test
One simple method used to determine if a graph is a relation or a function is the vertical line test.
If you draw a vertical line at any point on a coordinate plane, and it intersects the graph only once, it is a function
If it intersects the graph more than once, it is not a function
All functions are relations, but not all relations are functions

6. Is this a function?

7. Is this a function?

8. Is this a function?

9. Is this a function?

10. Is this a function?

11. Is this a function?

12. Domain and Range
The set of first elements of a relation is called the domain. When graphing a relation, the set of first elements will be the x-values of the graph.
The set of related second elements of a relation is called the range. When graphing a relation, the set of second elements will be the y-values of the graph.

13. Example: Arrow Diagram
Here are two relations that relate vehicles to the number of wheels each has: 1 2 3 4
Bicycle
Car
Motorcycle
Tricycle
Unicycle
unicycle
is the number of wheels on a has this number of wheels
  x y x y
This diagram does NOT represent a function because there is one element in the first set that is related with two elements in the second set
This diagram DOES represent a function because each element in the first set associates with exactly one element in the second set

14. Example: Ordered Pairs
{(1, unicycle), (2, bicycle), (2, motorcycle), (3, tricycle), (4, car)}
The set of ordered pairs above does not represent a function because two ordered pairs have the same first element.
The domain is the set of first elements: {1,2,3,4}
The range is the set of associated second elements: {unicycle, bicycle, motorcycle, tricycle, car}
{(unicycle, 1), (bicycle, 2), (motorcycle, 2), (tricycle, 3), (car, 4)}
The set of ordered pairs above represents a function because the ordered pairs have different first elements
The domain is the set of first elements: {unicycle, bicycle, motorcycle, tricycle, car}
The range is the set of associated second elements: {1,2,3,4}

15. Example: You do
With a partner, try the Check your understanding question on page 266 of the textbook

16. Example: Table of Values
In many jobs, a person’s pay, P dollars, often depends on the number of hours worked, h. Therefore, we say that P is the dependent variable. The number of hours worked, h, does NOT depend on the pay, P, so we say that h is the independent variable.
Domain
Hours worked, h
Gross Pay, P($) 1 12 2 24 3 36 4 48 5 60
Range
The values of the independent variable are listed first, and these elements belong to the domain.
The values of the dependent variable are listed second, and these elements belong to the range.

17. Example: You do
With a partner, try the Check your understanding question on page 267 of the textbook

18. Function Notation
Function notation is a way of showing the independent variable in a function; for example, f(x) means that the value of the function f depends on the value of the independent variable x.
In the last example, the value of the gross pay, P, depends on the value of the independent variable (hours worked), h, so in function notation it would be written as P(h).
We say these as “f of x” and “P of h”

19. Example: You do
With a partner, try the Check your understanding question on page of the textbook

20. Homework
Page , #4, 6, 9, 10, 12, 14-16, 18-19, 22.