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Question

If f(x)=|x2| and g(x)=f(f(x)), then g(x) for x>2 is

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Solution

We have, f(x)=2x,x<2=x2,x2
Thus, we have,
g(x)=f{f(x)}=2f,f<2
=f2,f2
i.e., g(x)=2(2x),2x<2x<2
=(2x)2,2x2x<2
=2(x2),x2<2x2
=(x2),2,x22x2
i.e., g(x)=x,0<x<2
=x,x0
=4x,2x<4
=x4,4x
i.e., g(x)=x,x0
=x,0<x<2
=4x,2x<4
=x4,4x
Hence, we have for x>2
g(x)=1,2<x<4
=1,4x
The derivative of g(x) does not exist at x=4.

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