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Chapter 1 Class 12 Relation and Functions
Serial order wise

Example 12 - Show that f(x) = { x+1, if x is odd ; x-1, if x is even

Example 12 - Chapter 1 Class 12 Relation and Functions - Part 2
Example 12 - Chapter 1 Class 12 Relation and Functions - Part 3 Example 12 - Chapter 1 Class 12 Relation and Functions - Part 4 Example 12 - Chapter 1 Class 12 Relation and Functions - Part 5


Transcript

Example 12 Show that f : N β†’ N, given by f(x) = {β–ˆ(π‘₯+1 , 𝑖𝑓 π‘₯ 𝑖𝑠 π‘œπ‘‘π‘‘@π‘₯βˆ’1, 𝑖𝑓 π‘₯ 𝑖𝑠 𝑒𝑣𝑒𝑛)─ is both one-one and onto. Check one-one There can be 3 cases x1 & x2 both are odd x1 & x2 both are even x1 is odd & x2 is even If x1 & x2 are both odd f(x1) = x1 + 1 f(x2) = x2 + 1 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Putting f(x1) = f(x2) x1 + 1 = x2 + 1 x1 = x2 If x1 & x2 are both are even f(x1) = x1 – 1 f(x2) = x2 – 1 If f(x1) = f(x2) x1 – 1 = x2 – 1 x1 = 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 If x1 is odd and x2 is even f(x1) = x1 + 1 f(x2) = x2 – 1 If f(x1) = f(x2) x1 + 1 = x2 – 1 x2 – x1 = 2 which is impossible as difference between even and odd number can never be even Hence, if f(x1) = f(x2) , Then x1 = x2 ∴ function f is one-one Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Check onto f(x) = {β–ˆ(π‘₯+1 , 𝑖𝑓 π‘₯ 𝑖𝑠 π‘œπ‘‘π‘‘@π‘₯βˆ’1, 𝑖𝑓 π‘₯ 𝑖𝑠 𝑒𝑣𝑒𝑛)─ Let f(x) = y , such that y ∈ N x = {β–ˆ(π‘¦βˆ’1 , 𝑖𝑓 𝑦 𝑖𝑠 𝑒𝑣𝑒𝑛@𝑦+1, 𝑖𝑓 𝑦 𝑖𝑠 π‘œπ‘‘π‘‘)─ If x is odd f(x) = x + 1 y = x + 1 y – 1 = x x = y – 1 If x is odd, y is even If x is even f(x) = x – 1 y = x – 1 y + 1 = x x = y + 1 If x is even, y is odd Hence, if y is a natural number, x will also be a natural number i.e. x ∈ N Thus, f is onto.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.