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If f(x) = f , which of the following is true for all values

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GMAT Club Legend
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If f(x) = f , which of the following is true for all values [#permalink] New post 30 Jul 2018, 09:47
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Question Stats:

53% (00:36) correct 46% (01:18) wrong based on 13 sessions
If \(\frac{1}{2}f(x) = f(\frac{x}{2})\) , 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) \(f(x) =\sqrt{x-4}\)
[Reveal] Spoiler: OA

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Re: If f(x) = f , which of the following is true for all values [#permalink] New post 11 Aug 2018, 13:28
This question was phrased in an odd way in my opinion, but what it's asking is which of the functions fulfills the constraint that 1/2f(x) = f(x/2)

Depending on your math background, you may now that scalars don't work the same way inside roots and exponents as they do outside, and scalars outside functions will affect constants inside function, which we don't want. If you don't have that math background, you can check with values. Here is a somewhat convoluted explanation without values.

We can quickly rule out any option with a constant, as the constant in 1/2f(x) will be affected by the division by 2, but the constant in f(x/2) will not.

Example: f(x) = 2x+2

1/2(2x+2) = x+1 != 2x/2 + 2 = f(x/2)

We can also rule out the square and the square root by noting that:

(x/2)^2 = x^2/4 != x^2/2

and

sqrt{x/2} != sqrt{x/4} = 1/2*sqrt{x}

In these cases, the constant scales with the square or the square root, which we don't want.

Simply put, we want a linear function without a constant, and the only option that fulfills that constraint is option B.

Or you can quickly hammer in the functions with the scalar inside and outside the function into your calculator. If you do it on this question, as 13x is an early option and only one option is correct, it would have saved you time. But it obviously depends on which order you pick the options in.
GMAT Club Legend
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Joined: 07 Jun 2014
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Re: If f(x) = f , which of the following is true for all values [#permalink] New post 12 Aug 2018, 07:19
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Explanation

The question is asking, “For which function is performing the function on x and THEN multiplying by \(\frac{1}{2}\) the equivalent of performing the function on \(\frac{1}{2}\) of x?”


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.
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Re: If f(x) = f , which of the following is true for all values   [#permalink] 12 Aug 2018, 07:19
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If f(x) = f , which of the following is true for all values

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