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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 Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Putting f (x1) = f (x2) 1/x1 = 1/x2 x2 = x1 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 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) Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 1/x1 = 1/x2 x2 = x1 Hence, if f(x1) = f(x2) , x1 = x2 ∴ 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 For y = 2 x = 1/2 So, x is not a natural number Hence, f is not onto

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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.