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

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Injective but not surjective.


f:N \rightarrow N given byf(x)=x^3  .


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 us check for the given function is injection, surjection and bijection.

Let's start with the injection test.

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

\begin{aligned} &f(x)=x^{3}, f(y)=y^{3} \\ &f(x)=f(y) \\ &x^{3}=y^{3} \\ &x=y \end{aligned}

So,f is an injection

Surjection test:

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



x=\sqrt[3]{y} which may not be in N

So,f is not a surjection.

Here f is injective but not surjective, so the f is not a bijective.

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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