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Misc 6 - Give examples of f, g such that gof is injective - Composite funcions and one-one onto

Misc 6 - Chapter 1 Class 12 Relation and Functions - Part 2

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Misc 6 Give examples of two functions f: N → Z and g: Z → Z such that gof is injective but g is not injective. (Hint : Consider f(x) = x and g(x) = |x|). Let f(x) = x and g(x) = |x| where f: N → Z and g: Z → Z g(x) = 𝑥﷯ = 𝑥 , 𝑥≥0 ﷮−𝑥 , 𝑥<0﷯﷯ Checking g(x) injective(one-one) For example: g(1) = 1﷯ = 1 g(– 1) = 1﷯ = 1 Checking gof(x) injective(one-one) f: N → Z & g: Z → Z f(x) = x and g(x) = |x| gof(x) = g(f(x)) = 𝑓(𝑥)﷯ = 𝑥﷯ = 𝑥 , 𝑥≥0 ﷮−𝑥 , 𝑥<0﷯﷯ Here, gof(x) : N → Z So, x is always natural number Hence 𝑥﷯ will always be a natural number So, gof(x) has a unique image ∴ gof(x) is injective

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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.