Ex 1.2, 2 (v) - Chapter 1 Class 12 Relation and Functions

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Ex 1.2 , 2 Check the injectivity and surjectivity of the following functions: (v) f: Z → Z given by f(x) = x3 f(x) = x3 Checking one-one (injective) f (x1) = (x1)3 f (x2) = (x2)3 Now, f (x1) = f (x2) (x1)3 = (x2)3 x1 = x2 Since if f (x1) = f (x2) , then x1 = x2 ∴ It is one-one (injective) Check onto (surjective) f(x) = x3 Let f(x) = y , such that y ∈ Z x3 = y x = 𝒚^(𝟏/𝟑) Here y is an integer i.e. y ∈ Z Let y = 2 x = 𝑦^(1/3) = 𝟐^(𝟏/𝟑) So, x is not an integer ∴ f is not onto (not surjective) Hence, function f is injective but not surjective.