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M10: Q21 - PS

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Re: M10: Q21 - PS [#permalink] New post 24 Apr 2013, 04:14
Expert's post
tirbah wrote:
Bunuel wrote:
snkrhed wrote:
the key to solving this problem is that if not given a restriction as in b... the function can go on forever right?


Below is a detailed solution of this problem. Hope it helps.

If function f(x) satisfies f(x) = f(x^2) for all x, which of the following must be true?
A. f(4) = f(2)f(2)
B. f(16) - f(-2) = 0
C. f(-2) + f(4) = 0
D. f(3) = 3f(3)
E. f(0) = 0

We are told that some function f(x) has following property f(x) = f(x^2) for all values of x. Note that we don't know the actual function, just this one property of it. For example for this function f(3)=f(3^2) --> f(3)=f(9), similarly: f(9)=f(81), so f(3)=f(9)=f(81)=....

Now, the question asks: which of the following MUST be true?

A. f(4)=f(2)*f(2): we know that f(2)=f(4), but it's not necessary f(2)=f(2)*f(2) to be true (it will be true if f(2)=1 or f(2)=0 but as we don't know the actual function we can not say for sure);

B. f(16) - f(-2) = 0: again f(-2)=f(4) =f(16)=... so f(16)-f(-2)=f(16)-f(16)=0 and thus this option is always true;

C. f(-2) + f(4) = 0: f(-2)=f(4), but it's not necessary f(4) + f(4)=2f(4)=0 to be true (it will be true only if f(4)=0, but again we don't know that for sure);

D. f(3)=3*f(3): is 3*f(3)-f(3)=0? is 2*f(3)=0? is f(3)=0? As we don't know the actual function we can not say for sure;

E. f(0)=0: And again as we don't know the actual function we can not say for sure.

Answer: B.


Hi Bunuel,

thanks for the explanation. I found this question to be very tough and I seem to be the only one not being able to understand this even after your explanation :(

I could understand why B is the answer and why not C and D. But I am still confused with A and E. I have tried to explain my thought process below -

We know that f(x) = f(x^2) means f(2) = f(4) = f(16) and so on.....
A. f(4)=f(2)*f(2)
LHS is f(4) which means (16)
RHS is [color=#ff0000]f(2)*f(2) means 4*4 means (16)[/color]
Hence LHS = RHS?
As I am writing this, it occurred to me that here we are multiplying to functions {f(2)*f(2)} and we don't really know if multiplying of two functions will actually result in the multiplication of those two numbers/integers? it could result in some other function as well? Am I thinking in the right direction?

Coming to option E - it says f(0)=0
we know that f(x) = f(x^2)
if x is 0...its square (infact any exponent) or function of its square should always result in 0. I can't think through what's wrong with E?

Could you please help me understand where I am going wrong. Many thanks for your help.

Kind regards


f(4)=f(16), not 16 and f(2)*f(2)=f(4)*f(4), not 4*4.

As for E: f(x)=f(x^2) does not necessarily means that f(0)=0.
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Re: M10: Q21 - PS [#permalink] New post 24 Apr 2013, 04:30
Bunuel wrote:
tirbah wrote:
Bunuel wrote:
the key to solving this problem is that if not given a restriction as in b... the function can go on forever right?


Below is a detailed solution of this problem. Hope it helps.

If function f(x) satisfies f(x) = f(x^2) for all x, which of the following must be true?
A. f(4) = f(2)f(2)
B. f(16) - f(-2) = 0
C. f(-2) + f(4) = 0
D. f(3) = 3f(3)
E. f(0) = 0

We are told that some function f(x) has following property f(x) = f(x^2) for all values of x. Note that we don't know the actual function, just this one property of it. For example for this function f(3)=f(3^2) --> f(3)=f(9), similarly: f(9)=f(81), so f(3)=f(9)=f(81)=....

Now, the question asks: which of the following MUST be true?

A. f(4)=f(2)*f(2): we know that f(2)=f(4), but it's not necessary f(2)=f(2)*f(2) to be true (it will be true if f(2)=1 or f(2)=0 but as we don't know the actual function we can not say for sure);

B. f(16) - f(-2) = 0: again f(-2)=f(4) =f(16)=... so f(16)-f(-2)=f(16)-f(16)=0 and thus this option is always true;

C. f(-2) + f(4) = 0: f(-2)=f(4), but it's not necessary f(4) + f(4)=2f(4)=0 to be true (it will be true only if f(4)=0, but again we don't know that for sure);

D. f(3)=3*f(3): is 3*f(3)-f(3)=0? is 2*f(3)=0? is f(3)=0? As we don't know the actual function we can not say for sure;

E. f(0)=0: And again as we don't know the actual function we can not say for sure.

Answer: B.


Hi Bunuel,

thanks for the explanation. I found this question to be very tough and I seem to be the only one not being able to understand this even after your explanation :(

I could understand why B is the answer and why not C and D. But I am still confused with A and E. I have tried to explain my thought process below -

We know that f(x) = f(x^2) means f(2) = f(4) = f(16) and so on.....
A. f(4)=f(2)*f(2)
LHS is f(4) which means (16)
RHS is [color=#ff0000]f(2)*f(2) means 4*4 means (16)[/color]
Hence LHS = RHS?
As I am writing this, it occurred to me that here we are multiplying to functions {f(2)*f(2)} and we don't really know if multiplying of two functions will actually result in the multiplication of those two numbers/integers? it could result in some other function as well? Am I thinking in the right direction?

Coming to option E - it says f(0)=0
we know that f(x) = f(x^2)
if x is 0...its square (infact any exponent) or function of its square should always result in 0. I can't think through what's wrong with E?

Could you please help me understand where I am going wrong. Many thanks for your help.

Kind regards


f(4)=f(16), not 16 and f(2)*f(2)=f(4)*f(4), not 4*4.

As for E: f(x)=f(x^2) does not necessarily means that f(0)=0.[/quote][/quote]

Oh yeah...silly me...I got that now....we are not given the value of the function so we cant presume what that function might be equal to.....thanks a ton for your reply :)
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Re: M10: Q21 - PS [#permalink] New post 11 Apr 2014, 09:42
I agree with vishnuns, Great explanation! made it look like a cakewalk! :D
thanks Bunuel
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Re: M10: Q21 - PS   [#permalink] 11 Apr 2014, 09:42
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