Question

Let $f:R\to R$ be function such that $f\left(x\right)=x^3+x^2+3x+\sin x.$ Then

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

Answer 1

The correct option is B $f$ is one-one and onto

$f\left(x\right)=x^3+x^2+3x+\sin x.$
$\Rightarrow f^{\prime }\left(x\right)=3x^2+2x+3+\cos x\ge 3x^2+2x+3-1$, since $-1\le \cos x\le 1$
$\Rightarrow f^{\prime }\left(x\right)\ge 3x^2+2x+2$
Now discriminant of above quadratic is $=2^2-4\cdot 3\cdot 2<0$
Hence $f^{\prime }\left(x\right)>0$ for all $x\in R$
Therefore f is one-one function, also every odd degree polynomial
has it's range R. Thus f is an onto function