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Algebra Level 3

Let A A be the range and B B be the domain of the function y = sin ⁡ − 1 x y=\sin^{-1} x . Then which of the following represents A ∩ B A\cap B ?

[ − π 2 , π 2 ] \left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right] [ − 1 , 1 ] [-1,1] [ − 1 , π 2 ] \left[-1,\dfrac{\pi}{2}\right] [ − π 2 , 1 ] \left[-\dfrac{\pi}{2},1\right]

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Nihar Mahajan by Nihar Mahajan , 20, India

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1 solution

Domain of arcsin ⁡ x \arcsin x is the range of sin ⁡ x \sin x that is [ − 1 , 1 ] \text{[} -1,1\text{]} .
Now, one would think that range of arcsin ⁡ x \arcsin x is the domain of sin ⁡ x \sin x which is real numbers , however that is not true. Since, for the inverse to exist, we need the function to be bijective we restrict the domain of sin ⁡ x \sin x to [ − π 2 , π 2 ] \text{[} \dfrac{-\pi}{2} , \dfrac{\pi}{2} \text{]} . Thus range of arcsin ⁡ x \arcsin x is [ − π 2 , π 2 ] \text{[} \dfrac{-\pi}{2} , \dfrac{\pi}{2} \text{]} .
The domain is just a subset of the range in this case and thus itself is the intersection .

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Perfect! +1

Nihar Mahajan - 5 years, 2 months ago

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