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Question

Assertion :If f(x)=sgn(x) and g(x)=x(1x2), then fog(x) and gof(x) are continuous everywhere Reason: fog=1,x(1,0)(1,)0,x{1,0,1}1,x(,1)(0,1) and gof(x)=0,xR

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Assertion is incorrect and Reason is correct
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Solution

The correct option is D Assertion is incorrect and Reason is correct
We have, f(x)=sgn(x)=1,x<00,x=01,x>0
and g(x)=x(1x2)
Now, fog(x)=⎪ ⎪⎪ ⎪1,x(1x2)<00,x(1x2)=01,x(1x2)>0
Solving the inequality
x(1x2)<x(x1)(x+1)>0x(1,0)(1,)
Thus, we have
fog(x)=1,x(1,0)(1,)0,x{1,0,1}1,x(,1)(1,0)
which is continuous everywhere except at x{1,0,1}
Also, gof(x)=f(1f2)=⎪ ⎪ ⎪⎪ ⎪ ⎪1[1(1)2],x<00(102),x=01(112),x>0
gof(x)=0,xR which is continuous everywhere.

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