f(x) f(y)

shashank.ism wrote:Let f(x) be a function satisfying f(x) f(y) = f(xy) for all real x, y. If f(2) = 4, then what is the value of f(1/2)?

a) 0
b) 1/4
c) 1/2
d) 1

The CA is B

f(x*1) = f(x)*f(1)
=> f(1) =1
f(2*1/2) = f(2)*f(1/2) =1

4*f(1/2) =1
f(1/2) =1/4

ajith, great approach,

maths is amazingly interesting.

keep it up.

sadullaevd wrote:ajith, great approach,

maths is amazingly interesting.

keep it up.

Sadullaevd This type of question is solved by this approach. You have to first find out what do you need to get to the answer step by step..
If you go step by step.. you will find you know the value of f(2) and you need the value of f(1/2)
f(x)f(y) = f(xy)
put both values 2 and 1/2.
so, f(2).f(1/2) = f(2x1/2) --> (4) f(1/2) =f(1) --> f(1/2) = (f(1))/4 .....(i)

thus eventually we need the value of f(1). Now for f(1) you will surely try some combination in the original eq. f(x)f(y)= f(xy)
and you only the value of f(2) as yet. so just go on and put 2,1 in the eq and see the magic.
f(2).f(1) = f(2x1) --> f(1) = (f(2))/(f(2)) = 1

so from (i), f(1/2) = f(1)/4 = 1/4 .

Ajith, i dint quite understand your explanation. Though i got the answer as B

Dont know how

bhumika.k.shah wrote:Ajith, i dint quite understand your explanation. Though i got the answer as B
Dont know how

As long as you get the answer, it doesnt matter, does it?
Jokes apart,

f(x*y) = f(x) *f(y) -----(1)

it is true for any number which can be expressed in x*y format
Now we take the advantage of the fact that x*1 = 1*x = x

f (x) = f(x*1), applying (1)
= f(x)*f(1)

f(x) = f(x)*f(1)
Which implies,
f(1) =1

[Because at least for value of x=2; f(x) is not zero]

Now x*1/x = 1
so is 2*1/2 = 1

f(1) = f(2*1/2) = f(2)*f(1/2) = 4*f(1/2)
1= 4*f(1/2)
f(1/2) = 1/4

Now, please tell me where is the problem and I will be more than happy to explain

bhumika.k.shah wrote:Ajith, i dint quite understand your explanation. Though i got the answer as B

Dont know how

Bhumika I think Ajith solution is loud and clear enough to be easily understood.. Just go through it again. Though he has calculated f(1) before getting eq. but it is not a problem.

Btw I have given a detailed solution with step - to - step approach. Go through it, Hope u understand it. otherwise there is certainly a problem and u need a lot of practice.