Question

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

• A. f is one-one but not onto R
• B. f is onto R but not one-one
• C. f is one-one and onto R
• D. f is neither one-one nor onto R

Answer 1

Answer: (C)

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