Example 23 - Chapter 1 Class 12 Relation and Functions (Term 1)
Last updated at Jan. 28, 2020 by
Last updated at Jan. 28, 2020 by
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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 Rough Checking inverse of f:X → Y Step 1: Calculate g: Y → X Step 2: Prove gof = IX Step 3: Prove fog = IY g is the inverse of f 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 Rough Checking inverse of f:X → Y Step 1: Calculate g: Y → X Step 2: Prove gof = IX Step 3: Prove fog = IY g is the inverse of f 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 Rough Checking inverse of f:X → Y Step 1: Calculate g: Y → X Step 2: Prove gof = IX Step 3: Prove fog = IY g is the inverse of f 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 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 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 Now, Checking for y = f(x) Putting value of x in f(x) f(x) = f((𝑦 − 3)/4) = 4((𝑦 − 3)/4) + 3 = y − 3 + 3 = y 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 such that f(x) = y Thus, f is onto Since f is one-one and onto f is invertible Finding inverse Inverse of x = 𝑓^(−1) (𝑦) = (𝑦 − 3)/4 ∴ Inverse of f = g(y) = (𝒚 − 𝟑)/𝟒 where g: Y → N
Finding Inverse
Inverse of a function
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Ex 1.3, 5 (i) Deleted for CBSE Board 2022 Exams
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Example 28 (a) Deleted for CBSE Board 2022 Exams
Misc 11 (i) Important Deleted for CBSE Board 2022 Exams
Ex 1.3, 11 Deleted for CBSE Board 2022 Exams
Example 27 Important Deleted for CBSE Board 2022 Exams
Misc 1 Deleted for CBSE Board 2022 Exams
Ex 1.3 , 6 Deleted for CBSE Board 2022 Exams
Ex 1.3, 14 (MCQ) Important Deleted for CBSE Board 2022 Exams
Example 23 Important Deleted for CBSE Board 2022 Exams You are here
Misc 2 Deleted for CBSE Board 2022 Exams
Ex 1.3 , 4 Deleted for CBSE Board 2022 Exams
Example 24 Deleted for CBSE Board 2022 Exams
Ex 1.3 , 8 Important Deleted for CBSE Board 2022 Exams
Example 25 Important Deleted for CBSE Board 2022 Exams
Ex 1.3 , 9 Important Deleted for CBSE Board 2022 Exams
Finding Inverse
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