Functions 15 2 CS708 1 Functions 15 2

Functions – (15 - 2) CS-708 1

Functions – (15 – 2 a) CS-708 2

Example – (15 - 3) CS-708 3

Solution – (15 – 3 a) CS-708 4

Solution – (15 – 3 b) CS-708 5

Solution – (15 – 3 c) CS-708 6

Function – (15 - 5) CS-708 7

Remarks – (15 – 5 a) CS-708 8

Arrow Diagram of a Function – (15 - 6) CS-708 9

Example – (15 – 6 a) CS-708 10

Example – (15 – 6 b) CS-708 11

Nonfunction – (15 - 7) CS-708 12

Range of a Function – (15 - 8) CS-708 13

Remarks – (15 – 8 a) CS-708 14

Exercise – (15 - 9) CS-708 15

Exercise – (15 – 9 a) CS-708 16

Solution – (15 – 9 b) CS-708 17

Graph of a Function – (11 - 11) CS-708 18

Graph of a Function – (15 – 11 a) CS-708 19

Graph of a Function – (15 – 11 b) CS-708 20

Vertical Line Test – (15 - 12) CS-708 21

Example – (15 - 13) CS-708 22

Solution – (15 – 13 a) CS-708 23

Solution – (15 – 13 b) CS-708 24

Exercise – (15 - 15) CS-708 25

Exercise – (15 – 15 a) CS-708 26

Exercise – (15 – 15 b) CS-708 27

Exercise – (15 - 17) CS-708 28

Exercise – (15 – 17 a) CS-708 29

Exercise – (15 – 17 b) CS-708 30

Exercise – (15 - 18) CS-708 31

Solution – (15 – 18 a) CS-708 32

Functions Not Well Defined – (15 - 20) CS-708 33

Solution – (15 – 20 a) CS-708 34

Solution – (15 – 20 b) CS-708 35

Exercise – (15 - 21) CS-708 36

Solution – (15 – 21 a) CS-708 37

Solution – (15 - 22) CS-708 38

Binary Operations – (15 - 28) CS-708 39

Example – (15 – 28 a) CS-708 40

Injective Function or One-To-One Function – (16 - 2) CS-708 41

Arrow Diagram – (16 – 2 a) CS-708 42

Function not One-to-One – (16 - 3) CS-708 43

Example – (16 - 4) CS-708 44

Example – (16 – 4 a) CS-708 45

Example – (16 – 4 b) CS-708 46

Example – (16 – 4 c) CS-708 47

Example – (16 – 4 d) CS-708 48

Example – (16 – 4 e) CS-708 49

Example – (16 - 8) CS-708 50

Example – (16 - 9) CS-708 51

Solution – (16 – 9 a) CS-708 52

Example – (16 - 10) CS-708 53

Graph of One-To-One Function – (16 11) CS-708 54

Example – (16 – 11 a) CS-708 55

Surjective Function or Onto Function – (16 - 12) CS-708 56

Subjective Function or Onto Function – (16 – 12 a) CS-708 57

Function not Onto – (16 – 13) CS-708 58

Example – (16 - 14) CS-708 59

Solution – (16 – 14 a) CS-708 60

Example – (16 – 15) CS-708 61

Solution – (16 - 16) CS-708 62

Solution – (16 – 16 a) CS-708 63

Example – (16 – 20) CS-708 64

Solution – (16 – 20 a) CS-708 65

Solution – (16 – 20 b) CS-708 66

Bijective Function – (16 - 24) CS-708 67

Example – (16 – 24 a) CS-708 68

Example – (16 - 25) CS-708 69

Graph of Bijective Function – (16 - 27) CS-708 70

Graph of Bijective Function – (16 – 27 a) CS-708 71

Identity Function – (16 - 28) CS-708 72

Example – (16 – 28 a) CS-708 73

Constant Function – (16 – 30) CS-708 74

Example – (16 – 30 a) CS-708 75

Equality of Functions – (17 - 2) CS-708 76

Example – (17 – 2 a) CS-708 77

Example – (17 – 2 b) CS-708 78

Example – (17 - 3) CS-708 79

Solution – (17 – 3 a) CS-708 80

Inverse of a Function – (17 - 4) CS-708 81

Inverse of Injective Function – (17 - 5) CS-708 82

Inverse of Surjective Function – (17 - 6) CS-708 83

Inverse of Bijective Function – (17 -7) CS-708 84

Inverse Function – (17 - 8) CS-708 85

Arrow Diagram – (17 – 8 a) CS-708 86

Example – (17 - 9) CS-708 87

Example – (17 - 11) CS-708 88

Example – (17 – 11 a) CS-708 89

Example – (17 – 11 b) CS-708 90