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Question

A function f:RR satisfies the equation f(x)f(y)f(xy)=x+y for all x,yR and f(1)>0 then

A
f(x)=x+12
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B
f(x)=x2+1
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C
f(x)=x2+1
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D
f(x)=x+1
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Solution

The correct option is D f(x)=x+1
Taking x=y=1, we get
f(1)f(1)f(1)=1+1(f(1))2f(1)2=0(f(1)2)(f(1)+1)=0f(1)=2(f(1)>0)
Taking y=1, we get
f(x)f(1)f(x)=x+12f(x)f(x)=x+1f(x)=x+1

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