Example 11 - Show f(x) = x2 is neither one-one nor onto - To prove injective/ surjective/ bijective (one-one & onto)

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  1. Chapter 1 Class 12 Relation and Functions
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Example 11 Show that the function f: R R, defined as f(x) = x2, is neither one-one nor onto f(x) = x2 Checking one-one f (x1) = (x1)2 f (x2) = (x2)2 f (x1) = f (x2) (x1)2 = (x2)2 x1 = x2 or x1 = x2 Since x1 does not have unique image, It is not one-one Eg: f( 1) = ( 1)2 = 1 f(1) = (1)2 = 1 Here, f( 1) = f(1) , but 1 1 Hence, it is not one-one Check onto f(x) = x2 Let f(x) = y , such that y R x2 = y x = Note that y is a real number, so it can be negative also Putting y = 3 x = ( 3) Hence, x is not real So, f is not onto

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