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Question
Give an example of a function which is neither one-one nor onto ?
Solution
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
⇒ x2 = 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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