#### Provide solution for RD Sharma maths class12 Chapter Functions exercise 2.1 question 4

$f:A \rightarrow A$ is neither one-one nor onto.

Given:

$A=\{-1,0,1\} \text { and } f=\{(x, x): x \in A\}$

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.

Solution:

We need to prove that$f:A \rightarrow A$ is neither one-one nor onto.

First of all we check the function with one-one,

\begin{aligned} &\begin{gathered} f(x)=x^{2} \\ \text { If } x=1, f(1)=1^{2}=1 \\ \text { If } x=-1, f(-1)=(-1)^{2}=1 \end{gathered}\\ &\text { Here } f(1) \text { and } f(-1) \text { have the same image } 1 \text { . } \end{aligned}

So,f is not one-one.

Now, we go for onto function,

Co-domain of $f=\left \{ 0,1 \right \}$

\begin{aligned} &f(x)=x^{2} \\ &f(1)=1^{2}=1 \\ &f(-1)=(-1)^{2}=1 \\ &f(0)=0 \end{aligned}

Range of $f =\left \{ -1,0.1 \right \}$

Codomain of$f\neq$ Range of $f$ . (both are not same)

Hence, f is not onto.