Question

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

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

Answer 1

Answer: (C)

$f(x)=x^3+5 x+1$, for all $x \in R$
$f^{\prime}(x)=3 x^2+5 >0$
$f(x)$ is montonic on $R$.
Also $f(x)$ being polynomial of degree and so range of $f(x)$ is $R$.
$\Rightarrow $ fis one-one and onto R.