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Home » Give an example of a function which is (i) neither one – one nor onto (ii) onto but not one – one.
Give an example of a function which is
(i) neither one – one nor onto
(ii) onto but not one – one.
CBSE | Class 12 | Exercise 2A | Functions | Maths | RS Aggarwal
Give an example of a function which is
(i) neither one – one nor onto
(ii) onto but not one – one.

Solution:

(i) Neither one-one nor onto
\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R} given by \mathrm{f}(\mathrm{x})=|\mathrm{x}|=\left\{\begin{array}{l}\mathrm{x}, \text { if } \mathrm{x} \geq 0 \\ -\mathrm{x}, \text { if } \mathrm{x} \leq 0\end{array}\right.
For One-one
\begin{array}{l} f(x)=f(y) \\ |x|=|y| \\ x=y \text { or } x=-y \end{array}
Therefore, it is not one-one.
For Onto
We know that \mathrm{f}(\mathrm{x})=|\mathrm{x}| is always non-negative. So, there won’t be any element in domain \mathrm{R} for which \mathrm{f}(\mathrm{x}) is negative.
Therefore, it is not onto.
Hence, \mathrm{f}(\mathrm{x})=|\mathrm{x}| is neither one-one nor onto.

(ii) Onto but not one-one
\mathrm{f}(\mathrm{x})=|\mathrm{x}| from the set of Real numbers to the Set of Whole numbers.
For one-one
\begin{array}{l} f(x)=f(y) \\ |x|=|y| \end{array}
\mathrm{x}=\mathrm{y} and -\mathrm{x}=\mathrm{y}(\mathrm{x} is a real number, so can be positive or negative)
Therefore, it is not one-one.
For onto
Every element is set of Real numbers will have a value in set of Whole numbers, as a result, it is onto.

i

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