# Example 27 - Chapter 1 Class 12 Relation and Functions

Last updated at May 29, 2018 by Teachoo

Last updated at May 29, 2018 by Teachoo

Transcript

Example 27 Consider f : {1, 2, 3} → {a, b, c} and g : {a, b, c} → {apple, ball, cat} defined as f (1) = a, f (2) = b, f (3) = c, g(a) = apple, g(b) = ball and g(c) = cat. Show that f, g and gof are invertible. Find out f –1, g –1 and (gof) –1 and show that (gof) –1 = f –1 o g –1. Checking for f f : {1, 2, 3} → {a, b, c} f (1) = a, f (2) = b, f (3) = c, f is invertible if it is one-one and onto Check one-one Since, all elements have unique image f is one-one Check onto Since, every image has a unique pre-image, ∴ f is onto Since f is one-one and onto, f is invertible Now, f = {(1, a), (2, b) , (3, c)} So, f-1 = {(a, 1), (b, 2), (c, 3)} Checking for g g : {a, b, c} → {apple , ball , cat} g(a) = apple, g(b) = ball , g(c) = cat, g is invertible if it is one-one and onto Check one-one Since, all elements have unique image g is one-one Check onto Since, every image has a unique pre-image g is onto Since g is one-one and onto g is invertible So, g = {(a, apple) , (b, ball) , (c, cat)} ∴ g–1 = {(apple, a), (ball, b), (cat, c)} Checking for gof So, gof will be gof = { (1, apple) , (2, ball) , (3, cat) } gof is invertible if it is one-one and onto Since gof is one-one and onto gof is invertible So, gof = { (1, apple) , (2, ball) , (3, cat) } ∴ (gof)–1 = {(apple, 1), (ball, 2), (cat, 3)} We need to show that (gof) –1 = f –1 o g –1 Finding f –1 o g –1 Hence, (gof)–1 = {(apple, 1), (ball, 2), (cat, 3)} f –1 o g –1= {(apple, 1), (ball, 2), (cat, 3)} Thus, (gof) –1 = f –1 o g –1 Hence proved

Examples

Example 1

Example 2

Example 3

Example 4 Important

Example 5

Example 6

Example 7

Example 8

Example 9

Example 10

Example 11 Important

Example 12 Important

Example 13 Important

Example 14

Example 15

Example 16

Example 17

Example 18 Important

Example 19 Important

Example 20

Example 21

Example 22

Example 23 Important

Example 24

Example 25 Important

Example 26

Example 27 You are here

Example 28 Important

Example 29

Example 30

Example 31

Example 32

Example 33

Example 34

Example 35

Example 36

Example 37

Example 38

Example 39

Example 40

Example 41

Example 42 Important

Example 43

Example 44

Example 45 Important

Example 46 Important

Example 47 Important

Example 48 Important

Example 49

Example 50

Example 51

Chapter 1 Class 12 Relation and Functions

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.