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Need solution for RD Sharma maths class 12 chapter Functions exercise 2.1 question 5 sub question (ii)

Answers (1)

Neither one-one nor onto.


            f:Z \rightarrow Z given by f(x)=x^2


One-one function means every element in the domain has a distinct image in the  co-domain.

Onto function means every element in the co-domain has at least one  pre image in the domain of function.

For any function to be a bijective, the given function be one-one and onto.


Let's start with the injection test.

Let x and y be any two elements in the domain (Z) such that f(x)=f(y)

  \begin{aligned} &f(x)=x^{2}, f(y)=y^{2} \\ &f(x)=f(y) \\ &x^{2}=y^{2} \\ &x=\pm y \end{aligned}

Therefore,f is not an injection

Surjection test:

Let y be any element in the codomain (Z) , such thatf(x)=y  for some element x in Z .



x=\pm \sqrt{y} which may not be in N


If   y=2, x=\pm \sqrt{2} which is not in N .

So,f is not a surjection.

The condition of bijection is as we know the function should be one-one and onto.

Hence for this, f is not bijection.

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