For all real x, let f(x)=x3+5 x+1, then

Question

For all real x, let f(x)=x3+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.

Tardigrade Admin · Accepted Answer

f(x)=x3+5 x+1, for all x ∈ R f prime(x)=3 x2+5 >0 f(x) is montonic on R. Also f(x) being polynomial of degree and so range of f(x) is R. ⇒ fis one-one and onto R.

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

$f$ is one-one but not onto on $R$ .

$f$ is onto on $R$ but not one-one.

fis one-one and onto on $R$ .

fis neither one-one nor onto on R.

$f (x) = x^{3} + 5 x + 1$ , for all $x \in R$
$f^{'} (x) = 3 x^{2} + 5 > 0$
$f (x)$ 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.

Q. For all real x, let f(x)=x3+5x+1, then

f is one-one but not onto on R.

f is onto on R but not one-one.

fis one-one and onto on R.