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M25-22

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M25-22 [#permalink]

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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. \(f(3) = 3f(3)\)
E. \(f(0) = 0\)

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Re M25-22 [#permalink]

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Official Solution:

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 the 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 \(f(3)=f(3^2)\) or \(f(3)=f(9)\), 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 cannot 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 cannot say for sure;

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


Answer: B
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Re: M25-22 [#permalink]

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Bunuel wrote:
Official Solution:

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 the 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 \(f(3)=f(3^2)\) or \(f(3)=f(9)\), 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 cannot 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 cannot say for sure;

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


Answer: B



Could you please explain as to why E is wrong
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Re: M25-22 [#permalink]

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Rahul2000 wrote:

Could you please explain as to why E is wrong

Hi Rahul2000, in my opinion the reason for it to be wrong is that it doesn't say: f(0) = f(0). It says f(0) = 0 which is not the same. Do you see the difference?
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Re: M25-22 [#permalink]

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New post 23 Oct 2014, 11:02
mdacaret wrote:
Rahul2000 wrote:

Could you please explain as to why E is wrong

Hi Rahul2000, in my opinion the reason for it to be wrong is that it doesn't say: f(0) = f(0). It says f(0) = 0 which is not the same. Do you see the difference?



Aaaah I see it now. So you mean F(X) could be say X+1 . Hence F(0) = F(0) would hold but not F(0) = 0.
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Re: M25-22 [#permalink]

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New post 07 Dec 2014, 12:34
Request you to please explain the answer with some other method or please explain more elaborately.
Most of these kind of function questions I am unable to understand.
Can you please suggest some study material for the same?
I have already studied MNAHATTAN Books.
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Re: M25-22 [#permalink]

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New post 08 Dec 2014, 05:13
manojpandey80 wrote:
Request you to please explain the answer with some other method or please explain more elaborately.
Most of these kind of function questions I am unable to understand.
Can you please suggest some study material for the same?
I have already studied MNAHATTAN Books.


Check function questions in our Special Questions Directory:

Operations/functions defining algebraic/arithmetic expressions
Symbols Representing Arithmetic Operation
Rounding Functions
Various Functions
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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: M25-22 [#permalink]

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New post 08 Dec 2014, 07:03
Sir,

I have gone through questions , you have given only answer keys but no explained solutions.
can you please give me links where i can read more about functions.
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Re: M25-22 [#permalink]

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Re M25-22 [#permalink]

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I think this is a high-quality question and the explanation isn't clear enough, please elaborate. I don't get how f(-2) = f(4) = f(16) are we squaring terms to get to this ?
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Re: M25-22 [#permalink]

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New post 20 Feb 2016, 21:13
This question was pretty tough. I guess since we don't know the actual function all we know is the F(x)=F(x^2) and this could go on infinitely. So F(x)=f(x^2)=F(x^2^2)=F(x^2^2^2) and so on. So basically any number equals the square of that number. So B is the only answer that works. I fell for the E trap though. :-(
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Re: M25-22 [#permalink]

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New post 26 Feb 2016, 15:16
Hard Question, fell for the trap.
So all you need to see is that F(-2) can become -16 and F(16) is 16.
Hope don't see this on test day.
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Re: M25-22 [#permalink]

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New post 28 Jul 2016, 06:35
We are given f(x) = f(x^2)

If possible you can construct an equation that meets this condition ... e.g [ x-1/1-x]

Simply plug in each answer choice and verify
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Re M25-22 [#permalink]

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New post 05 Aug 2016, 03:20
I think this is a high-quality question and I agree with explanation.
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Re: M25-22 [#permalink]

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New post 05 Aug 2016, 03:52
Hi,

I resolved this question assuming that a function that satisfies F(x)=F(X^2) for all x is a constant function. Am I correct?
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Re: M25-22 [#permalink]

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New post 11 May 2018, 12:46
So in B, f(-2) can be f(4) or f(16) but f(16) cannot be f(16^2). Why?
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Re: M25-22 [#permalink]

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New post 11 May 2018, 13:21
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Re: M25-22 [#permalink]

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New post 11 May 2018, 13:32
Bunuel wrote:
urvashis09 wrote:
So in B, f(-2) can be f(4) or f(16) but f(16) cannot be f(16^2). Why?


f(x) = f(x^2), so f(-2) = f(4) = f(16) = f(16^2) = f(256^2) = ...



Thank you, now I get it!
Re: M25-22   [#permalink] 11 May 2018, 13:32
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