Question

If $f:R\to R$ be a function such that $f\left(x\right)=x^3+x^2+4x+sinx.$ 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)

$f\left(x\right)=x^3+x^2+4x+sinx$
$\Rightarrow f^{{}^{\prime }}\left(x\right)=3x^2+2x+4+cosx$
$f^{{}^{\prime }}\left(x\right)=3\left[{\left(x+\frac{1}{3}\right)}^2+\frac{11}{9}\right]-\left(-cosx\right)>0$ $as3{\left(\left[{\left(x+\frac{1}{3}\right)}^2+\frac{11}{9}\right]\right)}_{min}=\frac{11}{3}$
and $-cosx$ has the maximum value $1.$
$\Rightarrow f\left(x\right)$ is strictly increasing and hence it is one-one
Also, $\underset{x\to \in fty}{\lim }f\left(x\right)\Rightarrow \in fty$ and $\underset{x\to -\in fty}{\lim }f\left(x\right)\Rightarrow -\in fty.$
Thus, the range of $f\left(x\right)$ is $R,$ hence it is onto.