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

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

Answers (1)

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 functionf(x) as iff(x_1)=f(x_2), thenx_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 functionf:N \rightarrow N , given byf(x)=x^2

Calculate f(x_1)andf(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 byf(x)=x^2is neither one-one nor onto.

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