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Question

Let f(xy)=f(x)f(y) for all x,yR. If f(1)=2 and f(4)=4, then f(4) equal to

A
4
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B
1
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C
12
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D
8
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Solution

The correct option is D 8
We have f(x,y)=f(x)f(y) for all x,yR.
Putting x=y=1, we get
f(1)=f(1)f(1)
f(1)[1f(1)]=0
f(1)=1 [f(1)0]
Now, f(1)=2
limh0f(1+h)f(1)h=2
f(1)limh0f(h)1h=2
limh0f(1)f(h)f(1)h=2 [using f(1)=1]
limh0f(h)1h=2 ....(i)
Now, f(4)=limh0f(4+h)f(4)h
=limh0f(4)f(h)f(4)h
={limh0f(h)1h}f(4)=2f(4) ....{from (i)}
=2×4=8

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