1. Algebra 2 Notes 2.1 – Relations and Functions

2. Relation(ship) Advice If you have a loved one think about how your relationship works… Normally if you put work into your relationship, you are going to get results out of it. (Hopefully the results you want and you do not break-up) Example: If the female cooks and cleans (input) for the male, hopefully the result (output) will be that the male will buy her jewelry and flowers.

3. Vocabulary Sheet

4. A Relation in Mathematics Relation

5. Definition A relation is a set of pairs of input and output values. Think about substituting some number in for the variable x in some algebraic equation. Input: number in for x Output: number you get after solving equation

6. Facts/Characteristics Multiple Ways to Represent a Relation: Ordered pairs: {(1,2), (2,4)…} Input/Output Table Points on Coordinate plane Mapping diagram Equation Sentence/Words

7. Examples of Relations {(4,7), (3,2), (0,0), (-1, -4)} XY 01 -24 37

8. Non-Examples of Relations 4 (constant) y (variable) 7z (expression)

9. Domain

10. Domain of a Relation Definition: The set of all inputs or x-coordinates

11. Facts/Characteristics Range will ALWAYS depend on the domain

12. Examples Relation: {(0, 3), (2,9), (-3,5)} Domain of this relation: {0, 2, -3} Relation: 0 1 Domain: {heart, star, smiley face}

13. Non-Examples Relation: {(0,1), (2,3), (4,5)} Domain is NOT: {1,3,5}

14. Range

15. Definition of Range The set of all outputs or y-coordinates

16. Facts/Characteristics Domain CodomainRange: {A,B} ABCABC Range is a subset of the codomain!

17. Examples Range: {Fred, Peter, James} Beth Jennifer Brittany Fred George Peter Eric James {(0,1),(1,1),(4,6),(8,-1)} Range: {-1,1,6} Relation:

18. Non-Examples Range is NOT {Beth, Jennifer, Brittany} Beth Jennifer Brittany Fred George Peter Eric James Relation: {(0,1),(1,1),(4,6),(8,-1)} Range is NOT: {0,1,4,8}

19. A Special Type of Relation(ship) Think about your loved one again… You SHOULDN’T be dating anyone else, so you should only be connected with your significant other – that’s the only way a RELATION(ship) FUNCTIONS Bob cannot FUNCTION correctly because he is connected to two girls at the same time. BOB Shirley April

20. FUNCTIONS Definition: a relation where every input has exactly one output Facts/Characteristics: “Vertical Line Test” – if a vertical line passes through the graph and touches it more than once, then the graph is NOT a function

21. Examples {(1,2), (2,2), (3,2), (4,2)} XY 10 21 30

22. Non-Examples InputsOutputs 1212 ABAB XY 12 13 15 110 125

23. Two different types of Functions One-to-One Functions Onto Functions

24. One-to-One Functions Definition: Every element of the range of the function corresponds to exactly one element of the domain Facts/Characteristics: Written 1-1 Horizontal Line Test: determines if a function is 1-1

25. Examples of 1-1 Function 12341234 ABCDEABCDE

26. Non-Examples of 1-1 Functions 12341234 XYZXYZ

27. Onto Functions Definition: each element of the range corresponds to an element of the domain Facts/Characteristics Also called Surjective function For all elements b in B there is an element a in A such that f(a)=b. Whole codomain will be the range!

28. Examples of Onto Functions ABCDABCD 123123

29. Non-Examples of Onto Functions ABCABC 12341234

30. What we need to know… Graphing a Relation Find domain and range of a relation Make a mapping diagram Identify if a relation is a function Function Notation

31. Graph the Relation {(-2,4), (3,-2), (-1,0), (1,5)}

32. Find the domain and range of the plotted points.

33. Make a Mapping Diagram For the relation {(-1,-2), (3, 6), (-5, -10), (3,2)}

34. Identifying Functions Determine whether the relation is a function. A)B) -2 0 5 3 4 0 2 3 3 5

35. Function Notation