Example 24 - Let f: N -> Y, f(n) = n^2. Show that f is invertible

Example 24 - Chapter 1 Class 12 Relation and Functions - Part 2
Example 24 - Chapter 1 Class 12 Relation and Functions - Part 3
Example 24 - Chapter 1 Class 12 Relation and Functions - Part 4
Example 24 - Chapter 1 Class 12 Relation and Functions - Part 5 Example 24 - Chapter 1 Class 12 Relation and Functions - Part 6 Example 24 - Chapter 1 Class 12 Relation and Functions - Part 7 Example 24 - Chapter 1 Class 12 Relation and Functions - Part 8

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Question 7 (Method 1) Let Y = {n2 : n ∈ N} ⊂ N. Consider f : N → Y as f (n) = n2. Show that f is invertible. Find the inverse of f f(n) = n2 Step 1 Put f(n) = y y = n2 n2 = y n = ± √𝑦 Since f : N → Y, n ∈ N, Rough Checking inverse of f:X → Y Step 1: Calculate g: Y → X Step 2: Prove gof = IX Step 3: Prove fog = IY So, n is positive ∴ n = √𝑦 Let g(y) = √𝑦 where g: Y → N Now, f(n) = n2 & g(y) = √𝑦 Step 2: gof = g(f(n)) = g(n2) = √((𝑛2)) = n Rough Checking inverse of f:X → Y Step 1: Calculate g: Y → X Step 2: Prove gof = IX Step 3: Prove fog = IY Hence, gof = n = IN Step 3: fog = f(g(y)) = f(√𝑦 ) = (√𝑦)2 = 𝑦^(1/2 × 2) = 𝑦^1 = y Hence, fog(y) = y = IY Rough Checking inverse of f:X → Y Step 1: Calculate g: Y → X Step 2: Prove gof = IX Step 3: Prove fog = IY Since gof = IN and fog = IY, f is invertible & Inverse of f = g(y) = √𝒚 Question 7 (Method 2) Let Y = {n2 : n ∈ N} ⊂ N. Consider f : N → Y as f (n) = n2. Show that f is invertible. Find the inverse of f f(n) = n2 f is invertible if it is one-one and onto Check one-one f(n1) = n12 f(n2) = n22 Put f(n1) = f(n2) n12 = n22 ⇒ n1 = n2 & n1 = – n2 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 As n ∈ N, it is positive So, n1 ≠ – n2 ∴ n1 = n2 So, if f(n1) = f(n2) , then n1 = n2 ∴ f is one-one Check onto f(n) = n2 Let f(x) = y , where y ∈ Y y = n2 n2 = y n = ± √𝑦 Since f : N → Y, n ∈ N, So, n is positive ∴ n = √𝑦 Now, Checking for y = f(n) Putting value of n in f(n) f(n) = f(√𝑦) = (√𝑦)^2 = 𝑦 For all values of y, y ∈ Y, There exists n ∈ N such that f(n) = y Hence, f is onto Since f(x) is one-one and onto, So, f(x) is invertible Finding inverse Inverse of x = 𝑓^(−1) (𝑦) = √𝑦 ∴ Inverse of f = g(y) = √𝒚

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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 14 years. He provides courses for Maths, Science and Computer Science at Teachoo