Relations and Functions

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 oneelement 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 is the number of wheels on a has this number of wheels • Here are two relations that relate vehicles to the number of wheels each has: ----------------------------------------------------------- 1 2 3 4 1 2 3 4 Bicycle Car Motorcycle Tricycle Unicycle Bicycle Car Motorcycle Tricycle unicycle 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

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 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 269-270 of the textbook

20. Homework • Page 270-273, #4, 6, 9, 10, 12, 14-16, 18-19, 22.