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Last updated at Jan. 28, 2020 by Teachoo

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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 Check one-one Since, all elements have unique image gof is one-one Check onto Since, every image has a unique pre-image gof is 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 f-1 = {(a, 1), (b, 2), (c, 3)} g–1 = {(apple, a), (ball, b), (cat, c)} 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

Finding Inverse

Identity Function

Inverse of a function

How to check if function has inverse?

Example 22

Ex 1.3, 5 Important

How to find Inverse?

Example 28

Misc 11 Important

Ex 1.3, 11

Example 27 Important You are here

Misc 1

Ex 1.3 , 6

Ex 1.3 , 14 Important

Example 23 Important

Misc 2

Ex 1.3 , 4

Example 24

Ex 1.3 , 8 Important

Example 25 Important

Ex 1.3 , 9 Important

Chapter 1 Class 12 Relation and Functions

Concept wise

- Relations - Definition
- Empty and Universal Relation
- To prove relation reflexive, transitive, symmetric and equivalent
- Finding number of relations
- Function - Definition
- To prove one-one & onto (injective, surjective, bijective)
- Composite functions
- Composite functions and one-one onto
- Finding Inverse
- Inverse of function: Proof questions
- Binary Operations - Definition
- Whether binary commutative/associative or not
- Binary operations: Identity element
- Binary operations: Inverse

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.