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Last updated at Jan. 28, 2020 by Teachoo

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Ex 1.2, 1 Show that the function f: R* → R* defined by f(x) = 1/x is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*? Solving for f: R* → R* f(x) = 1/x Checking one-one f (x1) = 1/x1 f (x2) = 1/x2 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 f (x1) = f (x2) 1/x1 = 1/x2 x2 = x1 x1 = x2 Hence, if f(x1) = f(x2) , x1 = x2 ∴ f is one-one Check onto f: R* → R* f(x) = 1/x Let y = f(x) , such that y ∈ R* y = 1/𝑥 x = 1/𝑦 Since y not equal to 0, x is possible Thus we can say that if y ∈ R – {0} , then x ∈ R – {0} also Now, Checking for y = f(x) Putting value of x in f(x) f(x) = f(1/𝑦) = 1/(1/y) = y Hence f is onto Thus, for every y ∈ R* , there exists x ∈ R* such that f(x) = y Hence, f is onto Show that the function f: R* → R* defined by f(x) = 1/x is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*? Now, domain R* is replaced by N , codomain remains R* Hence f : N → R* f(x) = 1/x Checking one-one f (x1) = 1/x1 f (x2) = 1/x2 f (x1) = f (x2) Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 1/x1 = 1/x2 x2 = x1 x1 = x2 Hence, if f(x1) = f(x2) , x1 = x2 ∴ f is one-one Check onto f: N → R* f(x) = 1/x Let y = f(x) , , such that y ∈ R* y = 1/𝑥 x = 1/𝑦 Since y is real number except 0, x cannot always be a natural number Example Let y = 2 x = 1/2 So, x is not a natural number Hence, f is not onto

To prove one-one & onto (injective, surjective, bijective)

One One function

Onto function

One One and Onto functions (Bijective functions)

Example 7

Example 8

Example 9

Example 11 Important

Misc 5

Ex 1.2, 5 Important

Ex 1.2 , 6

Example 10

Ex 1.2, 1 You are here

Ex 1.2, 12

Ex 1.2 , 2 Important

Ex 1.2 , 7

Ex 1.2 , 11

Example 12 Important

Ex 1.2 , 9

Ex 1.2 , 3

Ex 1.2 , 4

Example 50

Example 51 Important

Ex 1.2 , 10 Important

Misc. 4 Important

Example 13 Important

Example 14 Important

Ex 1.2 , 8 Important

Example 46 Important

Misc 10 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.