f(x) f(y)

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shashank.ism: Legendary Member; Posts: 1022; Joined: Mon Jul 20, 2009 11:49 pmFollowed by: 2 members; Location: Gandhinagar; Thanked: 41 times

by shashank.ism » Sat Feb 06, 2010 8:57 am

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

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Source: Beat The GMAT — Problem Solving | Sat Feb 06, 2010 8:57 am

ajith: Legendary Member; Posts: 1275; Joined: Thu Sep 21, 2006 11:13 pmFollowed by: 2 members; Location: Arabian Sea; Thanked: 125 times

by ajith » Sat Feb 06, 2010 9:16 am

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

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sadullaevd: Senior | Next Rank: 100 Posts; Posts: 81; Joined: Wed Apr 08, 2009 3:12 pm; Location: South Korea; Thanked: 4 times

by sadullaevd » Sun Feb 07, 2010 4:23 am

ajith, great approach,

maths is amazingly interesting.

keep it up.

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Think critically,
Assume nothing.

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shashank.ism: Legendary Member; Posts: 1022; Joined: Mon Jul 20, 2009 11:49 pmFollowed by: 2 members; Location: Gandhinagar; Thanked: 41 times

by shashank.ism » Sun Feb 07, 2010 5:46 am

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 .

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bhumika.k.shah: Legendary Member; Posts: 941; Joined: Sun Dec 27, 2009 12:28 amFollowed by: 1 member; Thanked: 20 times

by bhumika.k.shah » Sun Feb 07, 2010 6:51 am

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

Dont know how

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ajith: Legendary Member; Posts: 1275; Joined: Thu Sep 21, 2006 11:13 pmFollowed by: 2 members; Location: Arabian Sea; Thanked: 125 times

by ajith » Sun Feb 07, 2010 7:23 am

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

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shashank.ism: Legendary Member; Posts: 1022; Joined: Mon Jul 20, 2009 11:49 pmFollowed by: 2 members; Location: Gandhinagar; Thanked: 41 times

by shashank.ism » Sun Feb 07, 2010 8:35 am

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.

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