![Example 24 - Chapter 1 Class 12 Relation and Functions - Part 2](https://d1avenlh0i1xmr.cloudfront.net/40526c24-3c16-406f-8b9b-287ffd0f8938/slide58.jpg)
![Example 24 - Chapter 1 Class 12 Relation and Functions - Part 3](https://d1avenlh0i1xmr.cloudfront.net/cd32b395-2982-4a58-8cec-1848e1cbc0fd/slide59.jpg)
![Example 24 - Chapter 1 Class 12 Relation and Functions - Part 4](https://d1avenlh0i1xmr.cloudfront.net/893cc413-ab74-47da-b08c-30a766090457/slide60.jpg)
![Example 24 - Chapter 1 Class 12 Relation and Functions - Part 5](https://d1avenlh0i1xmr.cloudfront.net/f1b3d53a-9acd-4c9e-bdd0-790c30e42241/slide61.jpg)
![Example 24 - Chapter 1 Class 12 Relation and Functions - Part 6](https://d1avenlh0i1xmr.cloudfront.net/e5860b21-f0e7-4b0b-9cbc-7e1232d4ac76/slide62.jpg)
![Example 24 - Chapter 1 Class 12 Relation and Functions - Part 7](https://d1avenlh0i1xmr.cloudfront.net/d0a8bdba-60f7-4d80-acf0-a29ce6e32d2f/slide63.jpg)
![Example 24 - Chapter 1 Class 12 Relation and Functions - Part 8](https://d1avenlh0i1xmr.cloudfront.net/e17db9c7-c106-4274-ad2b-e19839b427b7/slide64.jpg)
Examples
Last updated at April 16, 2024 by Teachoo
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