MCQ
For real $x,$ let $f\left( x \right) = {x^3} + 5x + 1$,then
- A$f$ is one-one but not onto $R$
- B$f$ is onto $R$ but not one-one
- ✓$f$ is onto and one-one $R$
- D$f$ is neither onto nor one-one $R$
Now, $f^{\prime}(x)=3 x^{2}+5>0, \forall x \in R$
Thus, $f(x)$ is strictly increasing function.
So, $f(x)$ is one-one function.
Clearly, $f(x)$ is a continuous function and also increasing on $R.$
$\therefore $ $\mathop {\lim }\limits_{x \to - \infty } f(x) = - \infty $
and $\mathop {\lim }\limits_{x \to \infty } = \infty $
Hence, $f(x)$ takes every value between $-\infty$ and $\infty$.
Thus, $f(x)$ is onto function.
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