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Prove that the function f : R ->R, given...

Prove that the function `f : R ->R`, given by `f (x) = 2x`, is one-one and onto.

Text Solution

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(i) `f(x_(1))=f(x_(2)) rArr 2x_(1) =2x_(2) rArr x_(1) =x_(2).` So f is one-one
(ii) Let `y=2 x.` Then `x=(1)/(2)y`
Thus for each `y ` in codomain R there exists `(1)/(2)`y such that
`f((1)/(2)y) =(2 xx (1)/(2)) =y`
`:. ` f is onto.
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