Check sibling questions

Example 25 - Let f(x) = 4x^2 + 12x + 15. Show that f is invertible

Example 25 - Chapter 1 Class 12 Relation and Functions - Part 2
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 3
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 4
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 5
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 6
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 7
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 8
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 9
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 10
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 11
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 12
Example 25 - Chapter 1 Class 12 Relation and Functions - Part 13

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Transcript

Example 25 (Method 1) Let f : N → R be a function defined as f (x) = 4x2 + 12x + 15. Show that f : N→ S, where, S is the range of f, is invertible. Find the inverse of f. f(x) = 4x2 + 12x + 15 Step 1: Let f(x) = y y = 4x2 + 12x + 15 0 = 4x2 + 12x + 15 – y 4x2 + 12x + 15 – y = 0 4x2 + 12x + (15 – y) = 0 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 Comparing equation with ax2 + bx + c = 0 a = 4, b = 12 , c = 15 – y x = (−𝑏 ± √(𝑏^2 − 4𝑎𝑐))/2𝑎 Putting values x = (− 12 ± √(〖12〗^2 − 4(4) (15 − 𝑦) ))/2(4) x = (− 12 ± √(144 − 16(15 − 𝑦) ))/8 = (− 12 ± √(16(9 −(15 − 𝑦)))/8 = (− 12 ± √(16(9 −15 + 𝑦)))/8 = (− 12 ± √(16(𝑦 − 6)))/8 = (− 12 ± √16 √(𝑦 − 6))/8 = (− 12 ± √(4^2 ) √(𝑦 − 6))/8 = (− 12 ± 4√(𝑦 − 6))/8 = 4[− 3 ± √(𝑦 − 6)]/8 = (− 3 ± √(𝑦 − 6))/2 So, x = (− 3 + √(𝑦 − 6))/2 or (− 3 − √(𝑦 − 6))/2 As x ∈ N , So, x is a positive real number x cannot be equal to (− 3 − √(𝑦 − 6))/2 Hence, x = (− 𝟑 + √(𝒚 − 𝟔))/𝟐 Let g(y) = (− 3 + √(𝑦 − 6))/2 where g: S → N 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 2: gof = g(f(x)) = g(4x2 + 12x + 15) = (−3 + √(4𝑥^2 + 12𝑥 + 15 − 6))/2 = (−3 + √(4𝑥^2 +12𝑥 + 9))/2 = (−3 + √(〖(2𝑥)〗^2+ 3^2 +2(2𝑥) ×3))/2 = (−3 + √((2𝑥 + 3)^2 ))/2 = (−3 + 2𝑥 +3)/2 = 2𝑥/2 = x Hence, gof = 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(x)) = f((− 3 + √(𝑦 − 6))/2) = 4((− 3 + √(𝑦 − 6))/2)^2 + 12((− 3 + √(𝑦 − 6))/2) + 15 = 4(−3 + √(𝑦 − 6))^2/4 + 6(−3 + √(𝑦 −6)) + 15 = (−3 + √(𝑦 −6))^2– 18 + 6√(𝑦 −6) + 15 = (–3)2 + (√(𝑦 −6))^2– 6√(𝑦 −6) – 18 + 6√((6 + 𝑦) ) + 15 = 9 + y – 6 – 18 + 15 = y Hence, fog = y = IS Since, gof = IN & fog = IS f is invertible, and inverse of f = g(y) = (− 𝟑 + √(𝒚 − 𝟔))/𝟐 Example 25 (Method 2) Let f : N → R be a function defined as f (x) = 4x2 + 12x + 15. Show that f : N→ S, where, S is the range of f, is invertible. Find the inverse of f. f(x) = 4x2 + 12x + 15 f is invertible if it is one-one and onto Checking one-one f (x1) = 4(x1)2 + 12x1 + 15 f (x2) = 4(x2)2 + 12x2 + 15 Putting f(x1) = f (x2) 4(x1)2 + 12x1 + 15 = 4(x2)2 + 12x2 + 15 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 4(x1)2 – 4(x2)2 + 12x1 – 12x2 = 15 – 15 4(x1)2 – 4(x2)2 + 12x1 – 12x2 = 0 4[(x1)2 – (x2)2 ]+ 12[x1 – x2] = 0 4[(x1 – x2) (x1 + x2) ]+ 12[x1 – x2] = 0 4(x1 – x2) [(x1 + x2) + 3] = 0 (x1 – x2) [x1 + x2 + 3] = 0/4 (x1 – x2) [x1 + x2 + 3] = 0 (x1 – x2) = 0 ∴ x1 = x2 (x1 + x2 + 3) = 0 x1 = – x2 – 3 Since f: N → S So x ∈ N i.e. x is always positive, Hence x1 = – x2 – 3 is not true Hence, if f (x1) = f (x2) , then x1 = x2 ∴ f is one-one Check onto f(x) = 4x2 + 12x + 15 Let f(x) = y such that y ∈ S Putting in equation y = 4x2 + 12x + 15 0 = 4x2 + 12x + 15 – y 4x2 + 12x + 15 – y = 0 4x2 + 12x + (15 – y) = 0 Comparing equation with ax2 + bx + c = 0 a = 4, b = 12 , c = 15 – y x = (−𝑏 ± √(𝑏^2 − 4𝑎𝑐))/2𝑎 Putting values x = (− 12 ± √(〖12〗^2 − 4(4) (15 − 𝑦) ))/2(4) x = (− 12 ± √(144 − 16(15 − 𝑦) ))/8 = (− 12 ± √(16(9 −(15 − 𝑦)))/8 = (− 12 ± √(16(9 −15 + 𝑦)))/8 = (− 12 ± √(16(𝑦 − 6)))/8 = (− 12 ± √16 √(𝑦 − 6))/8 = (− 12 ± √(4^2 ) √(𝑦 − 6))/8 = (− 12 ± 4√(𝑦 − 6))/8 = 4[− 3 ± √(𝑦 − 6)]/8 = (− 3 ± √(𝑦 − 6))/2 So, x = (− 3 + √(𝑦 − 6))/2 or (− 3 − √(𝑦 − 6))/2 As x ∈ N , So, x is a positive real number x cannot be equal to (− 3 − √(𝑦 − 6))/2 Hence, x = (− 𝟑 + √(𝒚 − 𝟔))/𝟐 Now, Checking for y = f(x) Putting value of x in f(x) f(x) = f ((− 𝟑 + √(𝒚 − 𝟔))/𝟐) = 4((− 𝟑 + √(𝒚 − 𝟔))/𝟐)^2 + 12 ((− 𝟑 + √(𝒚 − 𝟔))/𝟐) + 15 = (−3+√(𝑦 − 6))^2 + 6 (−3+√(𝑦 − 6)) + 15 = 9+(y−6)−6√(𝑦 − 6) −18+6√(𝑦 − 6)+15 = 𝑦+24−24 = 𝑦 Thus, For every y in range of f, there is a pre-image x in N such that f(x) = y Hence, f is onto Since f(x) is one-one and onto, So, f(x) is invertible Calculating inverse Inverse of x = 𝑓^(−1) (𝑦) = (− 𝟑 + √(𝒚 − 𝟔))/𝟐

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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.