Ex 1.2, 10 - f(x) = (x-2/x-3). Is f one-one onto - Class 12 - To prove injective/ surjective/ bijective (one-one & onto)

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Ex 1.2 , 10 Let A = R {3} and B = R {1}. Consider the function f: A B defined by f (x) = x 2 x 3 Is f one-one and onto? Justify your answer. f (x) = x 2 x 3 Check one-one f (x1) = x1 2 x1 3 f (x2) = x2 2 x2 3 Putting f (x1) = f (x2) x1 2 x1 3 = x2 2 x2 3 (x1 2) (x2 3) = (x1 3) (x2 2) x1 (x2 3) 2 (x2 3) = x1 (x2 2) 3 (x2 2) x1 x2 3x1 2x2 + 6 = x1 x2 2x1 3x2 + 6 3x1 2x2 = 2x1 3x2 3x2 2x2 = 2x1 + 3x1 x1 = x2 Hence, if f (x1) = f (x2), then x1 = x2 f is one-one Check onto f (x) = x 2 x 3 Let f(x) = y such that y B i.e. y R {1} So, y = x 2 x 3 y(x 3) = x 2 xy 3y = x 2 xy x = 3y 2 x (y 1) = 3y 2 x = 3y 2 y 1 For y = 1 , x is not defined But it is given that y R {1} Hence , x = 3y 2 y 1 R {3} Hence f is onto

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.