Question

If $f:R \rightarrow R$ be a function such that $f\left(x\right)=x^{3}+x^{2}+4x+sin x.$ Then, the function $f\left(x\right)$ is

• A. one-one and onto
• B. one-one and into
• C. many-one and onto
• D. many-one and into

Answer 1

Answer: (A)

$f\left(x\right)=x^{3}+x^{2}+4x+sin x$
$\Rightarrow f^{'}\left(x\right)=3x^{2}+2x+4+cos x$
$f^{'}\left(x\right)=3\left[\left(x + \frac{1}{3}\right)^{2} + \frac{11}{9}\right]-\left(- cos x\right)>0$ $as \, 3 \, \left(\left[\left(x + \frac{1}{3}\right)^{2} + \frac{11}{9}\right]\right)_{min}=\frac{11}{3}$
and $-cos x$ has the maximum value $1.$
$\Rightarrow f\left(x\right)$ is strictly increasing and hence it is one-one
Also, $\underset{x \rightarrow \in fty}{l i m}f\left(x\right)\Rightarrow \in fty$ and $\underset{x \rightarrow - \in fty}{l i m}f\left(x\right)\Rightarrow -\in fty.$
Thus, the range of $f\left(x\right)$ is $R,$ hence it is onto.