Misc 7 - Give examples of f, g - gof is onto but f is not onto  - Miscellaneous

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  1. Chapter 1 Class 12 Relation and Functions
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Misc 7 Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto. (Hint: Consider f(x) = x + 1 and g (x) = 𝑥−1, 𝑥>1﷮&1, 𝑥=1﷯﷯) Let f: N → N be f(x) = x + 1 And, g: N → N be, g (x) = 𝑥−1, 𝑥>1﷮&1, 𝑥=1﷯﷯ We will first show that f is not onto. Checking f is not onto f: N → N be f(x) = x + 1 Let y = f(x), where y ∈ N So, y = x + 1 y – 1 = x x = y – 1 For y = 1, x = 1 – 1 = 0 But 0 is not a natural number ∴ f is not onto. Finding gof f(x) = x + 1 & g (x) = 𝑥−1, 𝑥>1﷮&1, 𝑥=1﷯﷯ So, gof = 𝑥, 𝑥>1﷮1, 𝑥=1﷯﷯ Let gof = y , where y ∈ N So, y = 𝑥, 𝑥>1﷮1, 𝑥=1﷯﷯ Here y is a natural number , As y = x So, x is also a natural number Hence, gof is onto.

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