1.

What is largest interval on which function sin^-1x is continuous a)[0,1] b)[-1,0] c)[-1,1] d)(-infinity,+infinity ) which one is correct ?​

Answer»

is the largest interval on which FUNCTION sin¯¹x is continuous ?

solution : y = sin¯¹x is inverse trigonometric function.

differentiating with RESPECT to x,

dy/dx = 1/√(1 - x²)

we see function will be differentiable in the interval (-1, 1) because there is no point between-1 and 1 where dy/dx is undefined.

now we KNOW, if a function is differentiable in (a, b) then function is continuous in [a, b]

so function is continuous in [-1, 1]

2nd METHOD : F(x) = sin¯¹x

⇒x = f(y) = siny

range of f(y) = domain of f(x)

we know, range of siny belongs to [-1, 1]

so, domain of f(x) = [-1, 1]

now you can apply limits for -1, 1 and 0.

you will see, f(-1⁺) = f(-1)

f(0¯) = f(0⁺) = f(0)

f(1¯) = f(1)

so function is continuous between -1 and 1

therefore option (c) is [-1, 1] is correct.


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