Question

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

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

Answer 1

Answer: (B)

Given that $f(x)=x^3+5x+1$
$\therefore f^{\prime }(x)=3x^2+5>0,\forall x\in R$
$\Rightarrow f(x)$ is strictly increasing on R
$\Rightarrow f(x)$ is one one
$\therefore$ Being a polynomial f (x) is continuous and increasing.
on R with $\underset{x\to \infty }{\lim }f(x)=-\infty$ and $\underset{x\to \infty }{\lim }f(x)=\infty$
$\therefore$ Range of $f=(-\infty ,\infty )=R$
Hence f is onto also. So, f is one one and onto R.