Q41 / 90 · JEE Main 2009 · Mathematics

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

  • A
    $f$ is one-one but not onto $R$
  • B
    $\mathrm{f}$ is onto $\mathrm{R}$ but not one-one
  • Correct answer
    C
    $f$ is one-one and onto $R$
  • D
    $f$ is neither one-one nor onto $R$

Solution

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

Hence, the correct answer is option: 'C'.

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