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Question

If f(x) and g(x) are two functions with g(x)=x1x and fg(x)=x31x3, then f(x) is equal to

A
3x2+3
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B
x21x2
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C
1+1x2
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D
3x2+3x4
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Solution

The correct option is A 3x2+3
fg(x)=x31x3
Writing x31x3 using (ab)3=a3b33ab(ab), we have
(x1x)3=x31x33x1x(x1x)
x31x3=(x1x)3+3(x1x)
We have,
f(g(x))=x31x3=(x1x)3+3(x1x)
As g(x)=x1x, this yields
f(x1x)=(x1x)3+3(x1x)
On putting x1x=t, we get
f(t)=t3+3t
Thus, f(x)=x3+3x
and f(x)=3x2+3

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