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Joined: 02 Sep 2009
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For which of the following functions does f(−x)=−f(x) for all real num  [#permalink]

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11 00:00

Difficulty:   55% (hard)

Question Stats: 62% (01:36) correct 38% (01:46) wrong based on 307 sessions

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For which of the following functions does f(−x)=−f(x) for all real number values of x?

A. x^8−x^4
B. x^2−x^6
C. x^5/x^7
D. x^9/(x^5+1)
E. x^5/(x^2+1)

Kudos for a correct solution.

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Re: For which of the following functions does f(−x)=−f(x) for all real num  [#permalink]

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9
Bunuel wrote:
For which of the following functions does f(−x)=−f(x) for all real number values of x?

A. x^8−x^4
B. x^2−x^6
C. x^5/x^7
D. x^9/(x^5+1)
E. x^5/(x^2+1)

Kudos for a correct solution.

A. f(-x) = (-x)^8 - (-x)^4 is not equal to -f(x) = -(x^8 - x^4).

B. option B has a similar structure to option A's answer choice!− f(-x) = (-x)^2 - (-x)^6 which is not equal to -f(x) = -(x^2 - x^6).

C. f(-x) = (-x)^5/(-x)^7 is also not equal to -f(x) = -(x^5/x^7). f(-x) yields a + answer, whilst -f(x) yields a − answer.

D. f(-x) = (-x)^9/((-x)^5+1) is not equal as -f(x) = (-1)*x^9/(x^5+1).

E. f(-x) = (-x)^5/((-x)^2+1) is equal to -f(x) = (-1)*x^5/(x^2+1), because (-x)^5 is equal to (-1)*x^5 and ((-x)^2+1) is the same as (x^2+1).

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Re: For which of the following functions does f(−x)=−f(x) for all real num  [#permalink]

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E = -X^5/(X^2+1)...Both the sides
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Re: For which of the following functions does f(−x)=−f(x) for all real num  [#permalink]

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Bunuel wrote:
For which of the following functions does f(−x)=−f(x) for all real number values of x?

A. x^8−x^4
B. x^2−x^6
C. x^5/x^7
D. x^9/(x^5+1)
E. x^5/(x^2+1)

Kudos for a correct solution.

If f(-x)=-f(x) then f(-x) + f(x) = 0

E) f(-x)= $$-x^5$$/x^2+1; f(x) is option E itself. Same denominator, subtract numerator you get 0.

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hich of the following functions does f(−x)=−f(x) for all real num  [#permalink]

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2
1
Answer = E. x^5/(x^2+1)

Change of sign happens does NOT happen when:

1. All powers are even

2. All powers are odd

If you observe the OA, only option E has the combination of odd power in numerator & even in denominator

$$f(-x) = \frac{(-x)^5}{(-x)^2+1} = \frac{-x^5}{x^2+1}$$

$$-f(x) = \frac{-x^5}{x^2+1}$$

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For which of the following functions does f(−x)=−f(x) for all real num  [#permalink]

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1
Use x = 2 as your guide to test the answer choices.

For choice A, f(−x) will be the same as f(x), because the even exponents ensure that whether you're taking 2 or -2 to the 8th or 4th powers the results will always be the same. Therefore, since f(−x)=f(x), then this doesn't satisfy the prompt "does f(−x)=−f(x).

Mathematically, using x = 2 you'll have f(−x)=(−2)8−(−2)4=256−16=230. Meanwhile, −f(x)=−(28−24)=−(256−16)=−230, so you don't have a match.

For choice B, the logic and math are pretty similar. Since both exponents are even, whether you're using a positive or negative value as your input, the results will be the same either way for f(2) and f(−2). So your positive/negative eyeball test should tell you that this one won't work either.

Mathematically, using x = 2, you'll have: f(−x)=(−2)2−(−2)6=4−64=−60. And −f(x)=−(22−26)=−(4−64)=−(−60)=60 They're not the same, so choice B doesn't hold.

Choice C is similar, in that x^5/x^7 is going to net out to 1x/^2. A negative input like 2 will still yield a positive output (1/4), whereas −f(x) is going to yield a negative number (f(x)=1/4, so −f(x)=−1/4).

By now you should see that choice D will be the same - with two odd exponents, the negative input will yield a negative divided by a negative, and a positive number overall. While taking −f(x) will yield a negative.

Mathematically, that's f(−x)=(−2)^9/(−2)^5+1=−29−31, which is a positive number, whereas −f(x)=−(2925+1), where the answer will clearly be negative.

Only choice E mixes an odd exponent with an even exponent, a necessary pairing to satisfy the question. Mathematically with 2, you'd have:

f(−x)=(−2)^5/(−2^)2+1=−32/5, and

−f(x)=−2^5/2^2+1=−32/5, for a match.

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Re: For which of the following functions does f(−x)=−f(x) for all real num  [#permalink]

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_________________ Re: For which of the following functions does f(−x)=−f(x) for all real num   [#permalink] 22 Aug 2018, 23:11
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