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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]  19 Aug 2016, 05:19
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82% (00:53) correct 17% (01:37) wrong based on 28 sessions
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) = $$x^2(x^3 - 1)$$

Practice Questions
Question: 13
Page: 152
[Reveal] Spoiler: OA

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Re: QOTD#11 Which of the following functions f defined for all n [#permalink]  19 Aug 2016, 05:24
Expert's post
Explanation

To determine which of the functions among the five choices has the property that f (–x) =–f (x) for all numbers x, you need to check each choice until you find one that has the property. In Choice A,

f(x) = $$\frac{x^3}{(x^2 +1)}$$ or

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

Therefore Choice A has the property f (– x) = –f (x), and since only one of the five choices can be the correct answer, the correct answer is Choice A.
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Re: QOTD#11 Which of the following functions f defined for all n [#permalink]  23 Aug 2016, 13:56
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sandy wrote:
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) = $$x^2(x^3 - 1)$$

Practice Questions
Question: 13
Page: 152

I prefer the solution that Sandy provided.
However, here's a different approach.

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]  10 Sep 2017, 09:07
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]  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]  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]  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]  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?

Cheers,
Brent
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Re: QOTD#11 Which of the following functions f defined for all n   [#permalink] 19 Jun 2018, 07:07
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