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KCET2021MathematicsFunctions

Let A= x: x R, x is not a positive integer ) Define f: A R as f(x)= 2 x x-1 , then f is

Options

  1. Ainjective but not surjective.
  2. Bsurjective but not injective.
  3. Cbijective.
  4. Dneither injective nor surjective.

Correct answer

A. injective but not surjective.

Step-by-step solution

Given function, f(x)= 2 x x-1 On differentiating w.r.t. x , we get aligned f^ (x) &= (x-1)(2)-2 x(1-0) (x-1)² &= 2 x-2-2 x (x-1)² = -2 (x-1)² < 0 aligned Function is strictly decreasing. Function is injective 2 x x-1 =y 2 x=y x-y y=x(y-2) Let x= y y-2 for y=2f is not surjective.

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