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\( f: R \rightarrow R: f(x)=x^{3} \) is (a) one-one and onto (b) one-one and into (c) many-one and onto (d) many-one and into |
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Answer» f(x) = x3 Let f(x1) = f(x2) ⇒ \(x_1^3=x_2^3\) ⇒ \(x_1^3-x_2^3=0\) ⇒ (x1 - x2) (\(x_1^2+x_2^2+x_1x_2\)) = 0 ⇒ (x1 - x2) = 0 (As \(x_1^2+x_2^2+x_1x_2\neq0\) except x1 = x2 = 0) ∴ Given function f(x) = x3 is one-one. Let y = x3 ⇒ x = y1/3 which is defined for real value of y. ∴ function f(x) = x3 is onto also. Hence, f(x) = x3 is one-one and onto. |
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