Question

If $f:\left[0,\frac{\pi }{2}\right]\to [0,\infty ]$ be a function defined by $y=\sin \left(\frac{x}{2}\right)$ , then $f$ is

• A. injective
• B. subjective
• C. objective
• D. None of these

Answer 1

Answer: (D)

We have, $y=\sin \frac{x}{2}$ and $0\le x\le \frac{\pi }{2}$
$\Rightarrow$ $0\le \frac{x}{2}\le \frac{\pi }{4}$
$\Rightarrow$ $0\le \sin \frac{x}{2}$
$\le \frac{1}{\sqrt{2}}$ $\Rightarrow$
$\left(0,\frac{1}{\sqrt{2}}\right)\subset [0,\infty )$
So, function is not surjective but function is injective as for any
$0\le x\le \frac{x}{2},\sin \frac{x}{2}$ gives unique image.