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Question

Which of the following is/are true?

A
If f:RR is a function satisfying f(xf(y))=f(f(y))+xf(y)+f(x)1 x,yR then f(x) can be 1x22.
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B
If f(x)=ax+bcxa, xac, a,b,cR{0}, then f(f(x))=x
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C
If f be a real valued integrable function satisfying f(x)+f(x+4)=f(x+2)+f(x+6) and g(x)=x+8xf(t)dt, xR, then g(x) can be a constant function.
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D
Domain of the function 1f(x) is [0,), where f3(x)=1x33xf(x), xR and f(1)1
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Solution

The correct options are
A If f:RR is a function satisfying f(xf(y))=f(f(y))+xf(y)+f(x)1 x,yR then f(x) can be 1x22.
B If f(x)=ax+bcxa, xac, a,b,cR{0}, then f(f(x))=x
C If f be a real valued integrable function satisfying f(x)+f(x+4)=f(x+2)+f(x+6) and g(x)=x+8xf(t)dt, xR, then g(x) can be a constant function.
D Domain of the function 1f(x) is [0,), where f3(x)=1x33xf(x), xR and f(1)1

For f(xf(y))=f(f(y))+xf(y)+f(x)1 x,yR
Put x=0, f(y)=0f(0)=1
Put x=a, f(y)=af(a)=1a22


For f(x)=ax+bcxa, f(f(x))=x


For f(x)+f(x+4)=f(x+2)+f(x+6)

xx+2
f(x)=f(x+8)
f(x) is periodic with period 8
x+8xf(t) dt=80f(t) dt=g(x)
g(x) is a constant function.


Let a=f(x),b=x,c=1
then a3+b3+c3=3abc
(a+b+c)×12[(ab)2+(bc)2+(ca)2]=0
f(1)1a+b+c=0f(x)=1x

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