Question

$f\left(x\right)=x-2\sin \frac{x}{2},f:R\to R$ is

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

Answer 1

The correct option is B one-one onto

$f\left(x\right)=x-2\sin \frac{x}{2}f:R\to R$

$\begin{array}{cc}{f}^{\prime }\left(x\right)& =1-\frac{2}{2}\cos \frac{x}{2}\\ f^{\prime }\left(x\right)& =1-\cos \frac{x}{2}\\ & =1-(1-2{\sin }^2\frac{x}{4})\\ & =1-1+2{\sin }^2\frac{x}{4}\end{array}$

$f^{\prime }\left(x\right)=2{\sin }^2\frac{x}{4}$

$\therefore f^{\prime }\left(x\right)$ will give only +ve values for all $x$

$\therefore f\left(x\right)$ is one -one function.

$f\left(x\right)=x-2\sin \frac{x}{2}$

we can clearly see that Range of

$f\left(x\right)$ will be $R$,

$\therefore$ co-domain of $f\left(x\right)=$ Range of $f\left(x\right)$

$\therefore f\left(x\right)$ is onto function.

Answer: option $\left(B\right)$.