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Given 1/2f(x)=f(x/2),

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Given 1/2f(x)=f(x/2), [#permalink] New post 17 Feb 2018, 12:03
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Given \(\frac{1}{2} f(x)= f (\frac{1}{2} * x)\), which of the following is true for all values of f(x)?

A.\(f(x)= 2x+2\)

B.\(f(x)=13x\)

C.\(f(x)=x^2\)

D.\(f(x)=x-10\)

E.\(f(x)=\sqrt{(x-4)}\)

I know the answer is B and I got it by trying each f(x) function. However the answer indicates, "the fastest method is to use logical: since the order of operations says that order does not matter with multiplication and division but does matter between multiplication and addition/subtraction, or multiplication and exponents, choose a function that has only multiplication and/or division."

Could anyone explain this logic? Thank you.
[Reveal] Spoiler: OA

Last edited by Carcass on 19 Feb 2018, 02:13, edited 2 times in total.
Edited by Carcass
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Re: A question on functions and formulas from Manhattan Prep 5lb [#permalink] New post 17 Feb 2018, 16:18
alihong2010 wrote:
Given 1/2f(x)=f(x/2), which of the following is true for all values of f(x)?

A.f(x)=2x+2
B.f(x)=13x
C.f(x)=x^2
D.f(x)=x-10
E.f(x)=sqrt(x-4)

I know the answer is B and I got it by trying each f(x) function. However the answer indicates, "the fastest method is to use logical: since the order of operations says that order does not matter with multiplication and division but does matter between multiplication and addition/subtraction, or multiplication and exponents, choose a function that has only multiplication and/or division."

Could anyone explain this logic? Thank you.


well, since the problem features transformations which feature multiplication (1/2*f(x)) and division (x/2), it's safe to say that the DM part of PEDMAS is used- I guess that leads to trying only equations with DM transformations? I'm not 100% sure either
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Re: A question on functions and formulas from Manhattan Prep 5lb [#permalink] New post 17 Feb 2018, 16:19
alihong2010 wrote:
Given 1/2f(x)=f(x/2), which of the following is true for all values of f(x)?

A.f(x)=2x+2
B.f(x)=13x
C.f(x)=x^2
D.f(x)=x-10
E.f(x)=sqrt(x-4)

I know the answer is B and I got it by trying each f(x) function. However the answer indicates, "the fastest method is to use logical: since the order of operations says that order does not matter with multiplication and division but does matter between multiplication and addition/subtraction, or multiplication and exponents, choose a function that has only multiplication and/or division."

Could anyone explain this logic? Thank you.


What page in the book was this featured, if you don't mind me asking?
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Re: Given 1/2f(x)=f(x/2), [#permalink] New post 19 Feb 2018, 02:14
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The fastest method is to use logic: since the order of operations says that order does not matter with multiplication and division but does matter between multiplication and addition/subtraction, or multiplication and exponents, choose a function that has only multiplication and/or division. Only answer choice (B) qualifies.


Hope this helps.
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Re: Given 1/2f(x)=f(x/2), [#permalink] New post 19 Feb 2018, 12:08
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Plugging might be a safe option if you are not familiar with polynomials.

The logical way mentioned here is:

given \(f(cx)=cf(x)\) where \(f(x)\) is a polynomial say \(f(x)= a_nx^n + a_{n-1}x^{n-1}.....+ a_1x + a_0\)

So \(f(cx)=a_n(cx)^n + a_{n-1}(cx)^{n-1}.....+ a_1cx + a_0\)

Here \(f(cx)=cf(x)\) if all coeffecients except \(a_1\) is 0 i.e a linear function or all coeffecients excepts \(a_0\) is 0 i.e. a constant function.

Since the options have only one linear function we can safely say that is the correct answer.
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Re: Given 1/2f(x)=f(x/2),   [#permalink] 19 Feb 2018, 12:08
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