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Give an Example of a Function Which is Neither One-one Nor onto ? - Mathematics

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प्रश्न

Give an example of a function which is neither one-one nor onto ?

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उत्तर

which is neither one-one nor onto.

f: Z → Z given by f(x) = 2x2 + 1

Injectivity:
Let x and y be any two elements in the domain (Z), such that f(x) = f(y).

f(x) = f(y)

⇒ 2x2+1 = 2y2+1

⇒ 2x2 = 2y2

⇒ x= y2

⇒ x = ± y

So, different elements of domain f may give the same image.
Thus, f is not one-one.

Surjectivity:
Let y be any element in the co-domain (Z), such that f(x) = y for some element x in Z(domain).

f(x) = y

⇒ 2x2+1=y

⇒ 2x2= y − 1

⇒  `x^2 = (y-1)/2`

⇒  `x = sqrt((y-1)/2)` ∉ Z always.

For example, if we take, y = 4,

 `x =± sqrt((y-1)/2) = ± sqrt((4-1)/2) = ±sqrt(3/2) ∉ Z `

So, x may not be in Z (domain).

Thus, f is not onto.

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पाठ 2: Functions - Exercise 2.1 [पृष्ठ ३१]

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आरडी शर्मा Mathematics [English] Class 12
पाठ 2 Functions
Exercise 2.1 | Q 1.3 | पृष्ठ ३१

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