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Question

If f:RR is continuous such that f(x+y)=f(x)+f(y),x,yR and f(1)=2 then f(100) equals to:

A
100
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B
50
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C
200
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D
0
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Solution

The correct option is C 200
From question, we know that:

f(x+y)=f(x)+f(y)

Let x=1,y=1
f(2)=2f(1)=4=2×2

f(4)=2f(2)=8=2×4 when (x=2,y=2)
f(8)=2f(4)=16=2×8 when (x=4,y=4)
f(16)=2f(8)=32=2×16 when (x=8,y=8)

Hence, it can be generalized that f(x)=2x

f(100)=2×100=200

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