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 $f '\left(x\right)=3x^{2}+5 > 0, ∀x ∈R$
$\therefore f \left(x\right)$ is strictly increasing function
$\therefore $ It is one-one
Clearly, $f\left(x\right)$ is a continuous function and also increasing on R,
$Lt_{x\rightarrow-\infty}\,f \left(x\right)=-\infty$ and $Lt_{x\rightarrow\infty}\,f \left(x\right)=\infty$
$\therefore f\left(x\right)$ takes every value between $−∞$ and $∞$ .
Thus, $f\left(x\right)$ is onto function.