1. 3.2 Relations And Functions

2. A relation is a set of ordered pairs. {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} This is a relation The domain is the set of all x values in the relation {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} The range is the set of all y values in the relation {(2,3), (-1,5), (4,-2), (9,9), (0,-6)} domain = {-1,0,2,4,9} These are the x values written in a set from smallest to largest range = {-6,-2,3,5,9} These are the y values written in a set from smallest to largest

3. Review A relation between two variables x and y is a set of ordered pairs An ordered pair consist of a x and y- coordinate – A relation may be viewed as ordered pairs, mapping design, table, equation, or written in sentences x-values are inputs, domain, independent variable y-values are outputs, range, dependent variable

4. Example 1 What is the domain ? {0, 1, 2, 3, 4, 5} What is the range ? {-5, -4, -3, -2, -1, 0}

5. Domain (set of all x ’ s) Range (set of all y ’ s) 1 2 3 4 5 2 10 8 6 4 A relation assigns the x ’ s with y ’ s This relation can be written {(1,6), (2,2), (3,4), (4,8), (5,10)}

6. Set A is the domain 1 2 3 4 5 Set B is the range 2 10 8 6 4 A function is a relation that has each input produce ONLY ONE output. The x value can only be assigned to one y This is a function ---it meets our conditions No x has more than one y assigned

7. Set A is the domain 1 2 3 4 5 Set B is the range 2 10 8 6 4 Must use all the x ’ s Let’s look at another relation and decide if it is a function. The x value can only be assigned to one y This is a function ---it meets our conditions No x has more than one y assigned The second condition says each x can have only one y, but it CAN be the same y as another x gets assigned to.

8. 1 2 3 4 5 2 10 8 6 4 Is the relation shown above a function? NO Why not??? 2 was assigned both 4 and 10

9. 3/9/2016 9:57 AM1-6 Relations and Functions9 Example 2 THIS IS NOT A FUNCTION!!! Input 4–509–1 –27 Output

10. 3/9/2016 9:57 AM1-6 Relations and Functions10 Example 3 YES Is this a function? Hint: Look only at the x-coordinates

11. 3/9/2016 9:57 AM1-6 Relations and Functions11 Example 4 Is this a function? Hint: Look only at the x-coordinates NO

12. 3/9/2016 9:57 AM1-6 Relations and Functions12 Choice One Choice Two Example 5 310310 –1 2 3 2 –1 3 2 3 –2 0 Which mapping represents a function? Choice 1

13. 3/9/2016 9:57 AM1-6 Relations and Functions13 Example 6 Which mapping represents a function? A.B. B

14. Example 7 Which situation represents a function? There is only one price for each different item on a certain date. The relation from items to price makes it a function. A fruit, such as an apple, from the domain would be associated with more than one color, such as red and green. The relation from types of fruits to their colors is not a function. a. The items in a store to their prices on a certain date b. Types of fruits to their colors

15. We commonly call functions by letters. Because function starts with f, it is a commonly used letter to refer to functions. This means the right hand side is a function called f This means the right hand side has the variable x in it The left side DOES NOT MEAN f times x like brackets usually do, it simply tells us what is on the right hand side.

16. So we have a function called f that has the variable x in it. Using function notation we could then ask the following: Find f (2). Remember---this tells you what is on the right hand side---it is not something you work. It says that the right hand side is the function f and it has x in it.

17. Find f (-2). This means to find the function f and instead of having an x in it, put a -2 in it. So let ’ s take the function above and make brackets everywhere the x was and in its place, put in a -2. Don ’ t forget order of operations---powers, then multiplication, finally addition & subtraction

18. 3/9/2016 9:57 AM1-6 Relations and Functions18 Function Notation Given g(x) = x 2 – 3, find g(-2). g(-2) = x 2 – 3 g(-2) = (-2) 2 – 3 g(-2) = 1

19. 3/9/2016 9:57 AM1-6 Relations and Functions19 For each function, evaluate f(0), f(1.5), f(-4), f(0) = f(1.5) = f(-4) = 3 4 4

20. 3/9/2016 9:57 AM1-6 Relations and Functions20 For each function, evaluate f(0), f(1.5), f(-4), f(0) = f(1.5) = f(-4) = 1 3 1

21. 3/9/2016 9:57 AM1-6 Relations and Functions21 For each function, evaluate f(0), f(1.5), f(-4), f(0) = f(1.5) = f(-4) = -5 1 1

22. Vertical Line Test Vertical Line Test: Tells you if a relation is a function when a vertical line drawn through its graph, passes through only one point. Take a pencil and move it from left to right (–x to x); if it crosses more than one point, it is not a function

23. 3/9/2016 9:57 AM1-6 Relations and Functions23 Vertical Line Test Would this graph be a function? YES

24. 3/9/2016 9:57 AM1-6 Relations and Functions24 Vertical Line Test Would this graph be a function? NO

25. 3/9/2016 9:57 AM1-6 Relations and Functions25 Is the following function discrete or continuous? What is the Domain? What is the Range? Discrete

26. 3/9/2016 9:57 AM1-6 Relations and Functions26 Is the following function discrete or continuous? What is the Domain? What is the Range? continuous

27. 3/9/2016 9:57 AM1-6 Relations and Functions27 Is the following function discrete or continuous? What is the Domain? What is the Range? continuous

28. 3/9/2016 9:57 AM1-6 Relations and Functions28 Is the following function discrete or continuous? What is the Domain? What is the Range? discrete