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

One-one but not onto.

Given:

$f:N \rightarrow N$ given by$f(x)=x^2$ .

Hint:

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.

Solution:

Let x and y be any two elements in the domain(N) 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=y \end{aligned}

Where x and y are in N , so we don’t get 1 value.

So,f is an injection.

Surjection test:

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

$f(x)=y$

$x^2=y$

$x=\pm \sqrt{y}$ which may not be in N

Example:

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.