Which of the following functions is surjectiv | vedclass

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(D) function $f: R 	o R$ is surjective if its range is $R$. $A$ polynomial of even degree with a positive leading coefficient has a range $[c, \infty

Vedclass · Accepted Answer

$f : R 	o R, f(x) = x^3 + 2x^2 - x + 1$

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(D) function $f: R 	o R$ is surjective if its range is $R$. $A$ polynomial of even degree with a positive leading coefficient has a range $[c, \infty)$,so it is not surjective. For $f(x) = x^3 + x + 1$,$f'(x) = 3x^2 + 1 > 0$,so it is strictly increasing,making it both injective and surjective (bijective). For $f(x) = \sqrt{1 + x^2}$,the range is $[1, \infty)$,so it is not surjective. For $f(x) = x^3 + 2x^2 - x + 1$,the degree is odd,so the range is $R$ (surjective). However,$f'(x) = 3x^2 + 4x - 1$. The discriminant $D = 16 - 4(3)(-1) = 28 > 0$,so $f'(x)$ changes sign,meaning the function is not monotonic and thus not injective. Therefore,$f(x) = x^3 + 2x^2 - x + 1$ is surjective but not injective.

Which of the following functions is surjective but not injective?

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