Question

The function $ f:X\to Y $ defined by $ f(x)=\sin x $ is one-one but not onto, if $ X $ and Y are respectively equal to

• A. R and R
• B. $ [0,\pi ] $ and [0,1]
• C. $ \left[ 0,\frac{\pi }{2} \right] $ and $ [-1,1] $
• D. $ \left[ \frac{-\pi }{2},\frac{\pi }{2} \right] $ and $ [-1,1] $

Answer 1

Answer: (C)

Since, $ f:X\to Y, $ then $ f(x)=sin\text{ }x $ Now, take option (c). Domain $ =\left[ 0,\frac{\pi }{2} \right], $ Range $ =[-1,1] $ For every value of $ x, $ we get unique value of y. But the value of y in $ [-1,0) $ does not have any pre-image. $ \therefore $ Function is one-one but not onto.