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The function f : R⟶R defined as (x) = x3 is :(a) One-on but not onto(b) Not one-one but onto(c) Neither one-one nor onto(d) One-one and onto |
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Answer» Option : (d) \(\text{let}\,f(x_1)=f(x_2)\forall x_1,x_2\in R\) \(\Rightarrow x^3_1=x_2^3\) \(\Rightarrow x^3_1-x_2^3=0\) \(\Rightarrow (x_1-x_2)(x_1^2+x_1x_2+x_2^2)=0\) \(\Rightarrow x_1=x_2\) \((\because x_1^2+x_1x_2+x_2^2 \neq0)\) \(\Rightarrow\) f is one - one \(\text{let}\,f(x)=x^3=y\;\forall y\in R\) \(\Rightarrow x=y^{\frac{1}{3}}\) Therefore, every image \(y\in R\) has a unique pre image \(y^{\frac{1}{3}}\) in \(R\). \(\Rightarrow f\) is onto \(\therefore\) f is one-one and onto. |
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