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

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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 thatf: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 off\neq Range of f . (both are not same)

Hence, f is not onto.

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