# Ex 1.3 , 6 - Chapter 1 Class 12 Relation and Functions

Last updated at Dec. 17, 2018 by Teachoo

Last updated at Dec. 17, 2018 by Teachoo

Transcript

Ex 1.3 , 6 Show that f: [โ1, 1] โ R, given by f(x) = ๐ฅ/(๐ฅ + 2) is one-one. Find the inverse of the function f: [โ1, 1] โ Range f. (Hint: For y โ Range f, y = f(x) = ๐ฅ/(๐ฅ + 2) , for some x in [โ1, 1], i.e., x = 2๐ฆ/(1 โ ๐ฆ) ) f(x) = x/(x+2) Check one-one f(x1) = ๐ฅ1/(๐ฅ1 + 2) f(x2) = ๐ฅ2/(๐ฅ2 + 2) Putting f(x1) = f(x2) ๐ฅ1/(๐ฅ1 + 2) = ๐ฅ2/(๐ฅ2 + 2) x1(x2 + 2) = x2(x1 + 2) x1x2 + 2x1 = x2x1 + 2x2 x1x2 โ x2x1 + 2x1 = 2x2 0 + 2x1 = 2x2 2x1 = 2x2 โ x1 = x2 Hence, if f(x1) = f(x2) , then x1 = x2 โด f is one-one Finding inverse f(x) = ๐ฅ/(๐ฅ + 2) Putting f(x) = y y = ๐ฅ/(๐ฅ + 2) y(x + 2) = x yx + 2y = x yx โ x = โ2y x(y โ 1) = โ2y x = (โ2๐ฆ )/(๐ฆ โ1) x = (โ2๐ฆ )/(โ1(โ๐ฆ + 1) ) x = (2๐ฆ )/((1 โ ๐ฆ) ) Let g(y) = (2๐ฆ )/((1 โ ๐ฆ) ) where g: R โ [โ1, 1], y โ 1 Thus, g is inverse of f Inverse of f = g(y) = (๐๐ )/((๐ โ ๐) ) , y โ 1

Finding Inverse

Identity Function

Inverse of a function

How to check if function has inverse?

Example 22

Ex 1.3, 5

How to find Inverse?

Example 28 Important

Misc 11

Ex 1.3, 11 Important

Example 27

Example 23 Important

Misc 1

Ex 1.3 , 6 Important You are here

Ex 1.3 , 14 Important

Misc 2

Ex 1.3 , 4

Example 24

Ex 1.3 , 8

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

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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.