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M10: Q21 - PS

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M10: Q21 - PS [#permalink] New post 22 Aug 2008, 14:37
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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) = 3*F(3)
(E) F(0) = 0

[Reveal] Spoiler: OA
B

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Between B and E, which one is correct and why?
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Re: M10: Q21 - PS [#permalink] New post 23 Aug 2008, 02:32
balboa wrote:
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) = 3*F(3)
e) F(0) = 0


Given : F(x) = F (x^2)
F(-2) = F (-2^2) = F (4)
F(4) = F(4^2) = F(16)

Therefore, F(16) - F(-2) = F (16) - F(16)=0

Ans is B
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Re: M10: Q21 - PS [#permalink] New post 23 Aug 2008, 22:58
You cant come to the conclusion that F(0) = 0 There is not information given about the vlaue of F(x).

Whats the OA?
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Re: M10: Q21 - PS [#permalink] New post 30 Aug 2008, 11:35
LoyalWater wrote:
balboa wrote:
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) = 3*F(3)
e) F(0) = 0


Given : F(x) = F (x^2)
F(-2) = F (-2^2) = F (4)
F(4) = F(4^2) = F(16)

Therefore, F(16) - F(-2) = F (16) - F(16)=0

Ans is B



I don't understand why F(-2) is taken to the 4th power....when the original function states that is to be taken to the 2nd power.
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Re: M10: Q21 - PS [#permalink] New post 01 Sep 2008, 00:15
Refer to the second post in this thread. The trick is here:

F(-2)=F(4)
and also
F(4)=F(16)
therefore
F(-2)=F(16)

I hope this helps.
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Re: M10: Q21 - PS [#permalink] New post 02 Sep 2008, 13:04
It does not provide information about the value of F(x) or F(x^2) .. so we can not perform any operation with F(x) or F(x^2).
The leaves us only F(16) - F(-2) = 0 and F(-2)+F(4) = 0 to be examined..

F(x) = F(x^2) => f(-2) = f(4) =f(16) .. so B is the answer.
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Re: M10: Q21 - PS [#permalink] New post 12 Apr 2010, 19:51
hear F(x)=F(x^2) but not any specific operation so,

F(x) = F(x^2) => f(-2) = f(4) =f(16) .. so B is the answer.
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Re: M10: Q21 - PS [#permalink] New post 14 Apr 2011, 05:50
F(-2) = F((-2)^2) = F(4)

F(4) = F(4^2) = F(16)

=> F(16) - F(-2) = 0

Answer - B
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Re: M10: Q21 - PS [#permalink] New post 16 Apr 2011, 13:50
Just by going the same logic why "C" cant be the answer?

As (-2)^2 = 4
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Re: M10: Q21 - PS [#permalink] New post 19 Apr 2011, 01:18
vijayahir wrote:
Just by going the same logic why "C" cant be the answer?

As (-2)^2 = 4


Because F(4) + f(4) can not be equal to zero unless F(4) = 0 which we have no information about, so the answer can not be C.
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Re: M10: Q21 - PS [#permalink] New post 21 Apr 2011, 16:57
balboa wrote:
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) = 3*F(3)
(E) F(0) = 0

[Reveal] Spoiler: OA
B

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Between B and E, which one is correct and why?


If for some reason the answer is B, why not A?
We are saying F(-2) = F(4) = F(16) and therefore F(16) - F(-2) => F(16) - F(16) =0 must be true.
So in the above example we are doing a simple arithmetic of 16 - 16 for F(16) - F(16).

Why not do 4 = 2 * 2 for F(4) = F(2)*F(2)? Can someone please help explain?
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Re: M10: Q21 - PS [#permalink] New post 21 Apr 2011, 23:36
mniyer wrote:
balboa wrote:
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) = 3*F(3)
(E) F(0) = 0

[Reveal] Spoiler: OA
B

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Between B and E, which one is correct and why?


If for some reason the answer is B, why not A?
We are saying F(-2) = F(4) = F(16) and therefore F(16) - F(-2) => F(16) - F(16) =0 must be true.
So in the above example we are doing a simple arithmetic of 16 - 16 for F(16) - F(16).

Why not do 4 = 2 * 2 for F(4) = F(2)*F(2)? Can someone please help explain?


We don't know what the function actually returns; we just know that the function returns the same value for a number and its square.

e.g..

f(2^{1/4})=100
f(2^{1/2})=100
f(2)=100
f(4)=100
f(16)=100

OR

f(2^{1/4})=2
f(2^{1/2})=2
f(2)=2
f(4)=2
f(16)=2

AND
f(3^{1/4})=200
f(3^{1/2})=200
f(3)=200
f(9)=200
f(81)=200

Now,
f(4) = f(2)*f(2)

If;
f(2)=0
f(4)=0
f(16)=0

f(2)*f(2)=0
f(4) = f(2)*f(2) as 0 = 0.

If,
f(2)=2
f(4)=2
f(16)=2
f(2)*f(2)=2*2=4

f(4) != f(2)*f(2) as 2 != 4

So, this function may be true but not MUST be true.
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Re: M10: Q21 - PS [#permalink] New post 08 May 2011, 11:48
What if one of the option is f(-2)-f(4)=0
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Re: M10: Q21 - PS [#permalink] New post 08 May 2011, 11:50
agdimple333 wrote:
What if one of the option is f(-2)-f(4)=0


This will also be correct.
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Re: M10: Q21 - PS [#permalink] New post 18 Apr 2012, 05:04
balboa wrote:
Do you mind telling me how E is incorrect?


Because there are many possibilities other than one given what if F(x) = 1 then F(0)=1
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Re: M10: Q21 - PS [#permalink] New post 19 Apr 2012, 13:31
the key to solving this problem is that if not given a restriction as in b... the function can go on forever right?
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Re: M10: Q21 - PS [#permalink] New post 19 Apr 2012, 13:37
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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.
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Re: M10: Q21 - PS [#permalink] New post 19 Apr 2013, 05:27
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.


Great explanation! I missed out on the initial part and got lost (f'x is the same as f'x2 the same as f'x4) - Great question!

Thanks & Regards,
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Re: M10: Q21 - PS [#permalink] New post 21 Apr 2013, 09:15
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When F(x) = F(x²) for all x, then F must be a constant function. That means F(0)=F(1)=F(...) whatever.
That is why B must be correct.
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Re: M10: Q21 - PS [#permalink] New post 23 Apr 2013, 09:03
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 f(2)*f(2) means 4*4 means (16)
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
Re: M10: Q21 - PS   [#permalink] 23 Apr 2013, 09:03
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