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Question

Let f(x)=sin(π6sin(π2sinx)) for all xR and g(x)=π2sinx for all xR. Let (fg)(x) denotes f(g(x)) and (gf)(x) denotes g(f(x)). Then which of the following is (are) true?

A
Range of f is [12,12]
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B
Range of fg is [12,12]
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C
limx0f(x)g(x)=π6
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D
There is an xR such that (gf)(x)=1
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Solution

The correct option is C limx0f(x)g(x)=π6
f(x)=sin(π6sin(π2sinx))
We know that
π2sinx[π2,π2]π6sin(π2sinx)[π6,π6]sin(π6sin(π2sinx))[12,12]
Therefore, the range of f is [12,12]

f(x)=sin(π6sin(π2sinx))g(x)=π2sinxfg(x)=sin(π6sin(π2sin(π2sinx)))
Similarly, we can solve
Thereore, the range of fg is [12,12]

limx0f(x)g(x)=limx0sin(π6sin(π2sinx))π2sinx=limx0sin(π6sin(π2sinx))π6sin(π2sinx)×π6sin(π2sinx)π2sinx=π6

gf(x)=1π2sin(sin(π6sin(π2sinx)))=1sin(π6sin(π2sinx))=sin12πsin(π6sin(π2sinx))>π6(sin123.14>sin112)f(x)>12 (π6=0.523)
Which is beyond the range of the function f(x), so its not possible.

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