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QOTD#11 Which of the following functions f defined for all n

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QOTD#11 Which of the following functions f defined for all n [#permalink] New post 19 Aug 2016, 05:19
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Which of the following functions f defined for all numbers x has the property that \(f(–x) = –f (x)\) for all numbers x ?

A. f(x)= \(\frac{x^3}{(x^2 +1)}\)

B. f(x)= \(\frac{(x^2 - 1)}{(x^2 + 1)}\)

C. f(x)= \(x^2(x^2 - 1)\)

D. f(x) = \(x(x^3 - 1)\)

E. f(x) = \(x^2(x^3 - 1)\)


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[Reveal] Spoiler: OA

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Re: QOTD#11 Which of the following functions f defined for all n [#permalink] New post 23 Aug 2016, 13:56
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Explanation

If f(–x) = –f(x) for all numbers x, then let's see what happens with some arbitrary value of x.

Let x = 2.
The correct answer will be such that f(-2) = -f(2)
A) f(-2) = -8/5 and -f(2) = -(8/5) = -8/5. In this case, f(–x) = –f(x) . KEEP
B) f(-2) = 3/5 and -f(2) = -(3/5) = -3/5. In this case, f(–x)–f(x) . ELIMINATE
C) f(-2) = 12 and -f(2) = -(12) = -12. In this case, f(–x)–f(x) . ELIMINATE
D) f(-2) = 18 and -f(2) = -(14) = -14. In this case, f(–x)–f(x) . ELIMINATE
E) f(-2) = -36 and -f(2) = -(28) = -28. In this case, f(–x)–f(x) . ELIMINATE

By the process of elimination, the correct answer is
[Reveal] Spoiler:
A


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Re: QOTD#11 Which of the following functions f defined for all n [#permalink] New post 10 Sep 2017, 09:07
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I think the question has not enough conditions to make Choice A unique. It assumes with all values of x, so what happens if x=0, if so, all Choices A, C, D, & E are correct.
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Re: QOTD#11 Which of the following functions f defined for all n [#permalink] New post 09 May 2018, 23:35
I think I'm a little confused with -f(x).

For instance, if f(x) = x+1, then does -f(x) = -(x+1) or -x+1
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Re: QOTD#11 Which of the following functions f defined for all n [#permalink] New post 10 May 2018, 12:20
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jen6 wrote:
I think I'm a little confused with -f(x).

For instance, if f(x) = x+1, then does -f(x) = -(x+1) or -x+1



If \(f(x) = x+1\) then \(-f(x) = -1 \times (x+1)\) and \(f(-x) = (-x+1)\).

If \(f\) is multiplied with -1 then everything gets multiplied with -1 if only x (variable) is multiplied with -1 then only variable terms get multiplied with -1.
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Re: QOTD#11 Which of the following functions f defined for all n [#permalink] New post 19 Jun 2018, 06:40
This may seem like a silly question, but how does (-x)^2+1 become x^2+1 in the denominator? What happens to the negative sign?
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Re: QOTD#11 Which of the following functions f defined for all n [#permalink] New post 19 Jun 2018, 07:07
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rnmcdaniel wrote:
This may seem like a silly question, but how does (-x)² + 1 become x² + 1 in the denominator? What happens to the negative sign?


Good question.

Let's think of -x as (-1)(x)

So, (-x)² = [(-1)(x)]²
= [(-1)(x)][(-1)(x)]
= (-1)(-1)(x)(x)
= (1)
= x²

So, (-x)² + 1 = x² + 1

Does that help?

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Brent
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Re: QOTD#11 Which of the following functions f defined for all n [#permalink] New post 14 Feb 2019, 01:21
dzungpham wrote:
I think the question has not enough conditions to make Choice A unique. It assumes with all values of x, so what happens if x=0, if so, all Choices A, C, D, & E are correct.


That is the thing!!

You see Choices A, C, D, & E are correct for only x = 0, and in some cases x=-1 but not for all values of x.

For example, if x = -2, then for choice B: f(-x) = 1/3 and -f(x) = -(1/3)

The only answer choice that fulfills the given condition for all values of x is only choice A.
Re: QOTD#11 Which of the following functions f defined for all n   [#permalink] 14 Feb 2019, 01:21
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