1. Class 12
2. Important Question for exams Class 12
3. Chapter 1 Class 12 Relation and Functions

Transcript

Example 23 (Method 1) Let f : N → Y be a function defined as f (x) = 4x + 3, where, Y = {y ∈ N: y = 4x + 3 for some x ∈ N}. Show that f is invertible. Find the inverse. Checking inverse Step 1 f(x) = 4x + 3 Let f(x) = y y = 4x + 3 y – 3 = 4x 4x = y – 3 x = 𝑦 − 3﷮4﷯ Let g(y) = 𝑦 − 3﷮4﷯ where g: Y → N Step 2: gof = g(f(x)) = g(4x + 3) = (4𝑥 + 3) − 3﷮4﷯ = 4𝑥 + 3 − 3﷮4﷯ = 4𝑥﷮4﷯ = x = IN Step 3: fog = f(g(y)) = f 𝑦 − 3﷮4﷯﷯ = 4 𝑦 − 3﷮4﷯﷯ + 3 = y – 3 + 3 = y + 0 = y = IY Since gof = IN and fog = IY, f is invertible & Inverse of f = g(y) = 𝒚 − 𝟑﷮𝟒﷯ Example 23 (Method 2) Let f : N → Y be a function defined as f (x) = 4x + 3, where, Y = {y ∈ N: y = 4x + 3 for some x ∈ N}. Show that f is invertible. Find the inverse. f is invertible if f is one-one and onto Checking one-one f(x1) = 4x1 + 3 f(x2) = 4x2 + 3 Putting f(x1) = f(x2) 4x1 + 3 = 4x2 + 3 4x1 = 4x2 x1 = x2 If f(x1) = f(x2) , then x1 = x2 ∴ f is one-one Checking onto f(x) = 4x + 3 Let f(x) = y, where y ∈ Y y = 4x + 3 y – 3 = 4x 4x = y – 3 x = 𝑦 − 3﷮4﷯ For every y in Y = {y ∈ N: y = 4x + 3 for some x ∈ N}. There is a value of x which is a natural number Thus, f is onto Since f is one-one and onto f is invertible Finding inverse f(x) = 4x + 3 For finding inverse, we put f(x) = y and find x in terms of y We have done that while proving onto y = f(x) y = 4x + 3 ⇒ x = 𝑦 − 3﷮4﷯ Let g(y) = 𝑦 − 3﷮4﷯ where g: Y → N ∴ Inverse of f = g(y) = 𝒚 − 𝟑﷮𝟒﷯

Chapter 1 Class 12 Relation and Functions

Class 12
Important Question for exams Class 12