#### Please solve RD Sharma class 12 Chapter Functions exercise 2.1 question 1 sub question (iii)  maths textbook solution.

The example for a function which is neither one-one nor onto.

Hint:

One-one function means every element in the domain has a distinct image in the co-domain. If one-one is given for any function$f(x)$ as if$f(x_1)=f(x_2)$, then$x_1=x_2$

Where, $x_1,x_2\in$ domain of $f(x)$

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

Solution:

Let function$f:N \rightarrow N$ , given by$f(x)=x^2$

Calculate $f(x_1)$and$f(x_2)$

\begin{aligned} &f\left(x_{1}\right)=x_{1}^{2} \\ &f\left(x_{2}\right)=x_{2}^{2} \\ &\left(x_{1}\right)^{2}=\left(x_{2}\right)^{2} \\ &x_{1}=x_{2} \text { or } x_{1}=-x_{2} \end{aligned}

Since,$x_1$ doesn’t have unique image,

Now,

$f(x)=x^2$

Let $f(x)=y$ such that $y\in R$

$x^2=y$

$x=\pm \sqrt{y}$

Since y is real number, then it can be negative also.

If $y=-2, x=\pm \sqrt{-2}$ which is not possible as the root of a negative number is not real.

Hence,x is not real, so f is not onto.

$\therefore$ Function $f:N \rightarrow N$ given by$f(x)=x^2$is neither one-one nor onto.