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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  1. Chapter 1 Class 12 Relation and Functions
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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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