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Question

Let f(x)=sin-1x and g(x)=x2-x-22x2-x-6. If g(2)=limx2g(x), then then the domain of the function fgis:


A

(-,-2][-43,)

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B

(-,-1][2,)

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C

(-,-2][-1,)

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D

(-,2][-32,)

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Solution

The correct option is A

(-,-2][-43,)


Explanation for Correct Answer

Step-1: Value of g(2) using limit:

f(x)=sin-1x

g(x)=x2-x-22x2-x-6

Finding the value of g(2)

g(2)=limx2g(x)=limx2x2-x-22x2-x-6=limx2x-2x+12x+3x-2=limx2x+12x+3=2+12×2+3=37

Therefore g(2)=37

Step-2 :Domain of composite function:

Calculating fg

fg=f(g(x))=sin-1(g(x))

Therefore,

-1g(x)1-1x+12x+31

Solving the first inequalities :

x+12x+31x+12x+3-10x+1-(2x+3)2x+30-x-22x+30x+22x+30

Using the

number line we get,

Similarly for the second inequality :

-1x+12x+30x+12x+3+10x+1+(2x+3)2x+303x+42x+3

Using the number line we get,

Therefore x is belongs to (-,-2][-43,)

Hence, option (A) is the correct answer.


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