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.