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NDA 1 2024 Maths, question 22

Topic: Algebra

Q22

Let A=A= {x∈R:−1<x<1x\in R:-1<x<1}. Which of the following is/are bijective functions from AA to itself ?

  1. f(x)=x∣x∣f(x)=x|x|

  2. g(x)=cos⁡(πx)g(x)=\cos (\pi x)

Select the correct answer using the code given below :

  1. (a) 1 onlyCorrect answer
  2. (b) 2 only
  3. (c) Both 1 and 2
  4. (d) Neither 1 nor 2

Solution.

Answer: (a) 1 only

  1. f(x)=x|x| is strictly increasing from −1 to 1 on the open interval and has inverse sign(y)√|y|, so it is bijective.
  2. But g(0)=1 is outside A; also g(x)=g(−x), so g is not injective.
  3. Only f qualifies.