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For real $x$, let $f(x)=x^{3}+5 x+1$, then

1) $f$ is one-one but not onto $R$
2) $f$ is onto $R$ but not one-one
3) $f$ is one-one and onto $R$
4) $f$ is neither one-one nor onto $R$
in Mathematics by Diamond (74,866 points) | 7 views

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(3)

Explanation

Given $f(x)=x^{3}+5 x+1$
Now $f(x)=3 x^{2}+5>0, \forall x \in R$
$\therefore f(x)$ is strictly increasing function
$\therefore$ It is one-one
Clearly, $f(x)$ is a continuous function and also increasing on $R$,
lt $f(x)=-\infty$ and $\quad$ lt $_{x \rightarrow \infty} f(x)=\infty$
$\therefore f(x)$ takes every value between $-\infty$ and $\infty$ Thus, $f(x)$ is onto function.

by Diamond (74,866 points)

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