Ex 1.2, 1 - Show f(x) = 1/x is one-one onto, where R* - To prove injective/ surjective/ bijective (one-one & onto)

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  1. Chapter 1 Class 12 Relation and Functions
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Ex 1.2, 1 Show that the function f: R* → R* defined by f(x) = 1﷮x﷯ is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*? Solving for f: R* → R* f(x) = 1﷮x﷯ Checking one-one f (x1) = 1﷮x1﷯ f (x2) = 1﷮x2﷯ f (x1) = f (x2) 1﷮x1﷯ = 1﷮x2﷯ x2 = x1 x1 = x2 Hence, if f(x1) = f(x2) , x1 = x2 ∴ f is one-one Check onto f: R* → R* f(x) = 1﷮x﷯ Let y = f(x) , such that y ∈ R* y = 1﷮𝑥﷯ x = 1﷮𝑦﷯ Since y not equal to 0, x is possible Thus we can say that if y ∈ R – {0} , then x ∈ R – {0} also Hence f is onto Show that the function f: R* → R* defined by f(x) = 1﷮x﷯ is one-one and onto, where R* is the set of all non-zero real numbers. Is the result true, if the domain R* is replaced by N with co-domain being same as R*? Now, domain R* is replaced by N , codomain remains R* Hence f : N → R* f(x) = 1﷮x﷯ Checking one-one f (x1) = 1﷮x1﷯ f (x2) = 1﷮x2﷯ f (x1) = f (x2) 1﷮x1﷯ = 1﷮x2﷯ x2 = x1 x1 = x2 Hence, if f(x1) = f(x2) , x1 = x2 ∴ f f is one-one Check onto f: N → R* f(x) = 1﷮x﷯ Let y = f(x) , , such that y ∈ R* y = 1﷮𝑥﷯ x = 1﷮𝑦﷯ Since y is real number except 0, x cannot always be a natural number Example: Let y = 2 x = 1﷮2﷯ So, x is not a natural number Hence, f is not onto

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