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Question

Let f:(0,)R be defined as f(x)=tan1(log2(x)) and g:RR be defined as g(x)=2x23x+22x2+3x+2

If fg denotes the composition of f and g, then

A
fg(1x)=fg(x)
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B
fg(x) is strictly increasing in the domain (,)
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C
fg(x) is many-one function
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D
Range of g(x) is [17,7]
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Solution

The correct options are
A fg(1x)=fg(x)
C fg(x) is many-one function
D Range of g(x) is [17,7]

fg(x)=tan1(log2(2x23x+22x2+3x+2))
Replacing x by 1x, we get
fg(1x)=tan1(log2(2x2+3x+22x23x+2))

Replacing x by x, we get
fg(x)=tan1(log2(2x2+3x+22x23x+2))
fg(1x)=fg(x)

As we already proved that fg(1x)=fg(x), so it can't be strictly increasing in the given domain (,).
So, fg(x) is many-one function.

Now, Let y=2x23x+22x2+3x+2
(2y2)x2+(3y+3)x+2y2=0
Now D0
(3y+3)24(2y2)(2y2)0
9y2+18y+916y2+32y160
7y2+50y70
7y250y+70

y[17,7]


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