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# f(x)

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Intern
Joined: 08 Jun 2009
Posts: 32

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14 Jun 2009, 00:15
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Question Stats:

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For which of the following functions f is f(x) = f(1-x) for all x?

(1) f(x) = 1-x
(2) f(x) = $$1-x^2$$
(3) f(x) = $$x^2-(1-x)^2$$
(4) f(x) = $$x^2(1-x)^2$$
(5) f(x) = $$x/(1-x)$$

[Reveal] Spoiler:
D

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Current Student
Joined: 03 Aug 2006
Posts: 112

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14 Jun 2009, 00:43
There are two ways to solve such questions.

One is the algebraic way and the other is number substitution. You can use the way that you are most comfortable with. We will try both here.

First the algebraic way:
The questions is asking for each of the given functions in the answer choices for which one does f(x) = f(1-x).

Lets look at the first choice
$$f(x)=1-x$$
$$f(1-x)=1-(1-x)=x$$
$$f(x)\neq f(1-x)$$

Lets look at the second choice
$$f(x)=1-x^2$$
$$f(1-x)=1-(1-x)^2=1-(1-2x+x^2)=2x+x^2$$
$$f(x)\neq f(1-x)$$

You can do this for each choice and you will see that (4) is the only choice where
$$f(x)=f(1-x)$$

Lets look at the fourth choice
$$f(x)=x^2(1-x)^2$$
$$f(x)=(1-x)^2(1-(1-x))^2=(1-x)^2(1-1+x)^2=(1-x)^2x^2$$
$$f(x)=f(1-x)$$

The other approach is the number substitution
Lets say x=2 then 1-x=2-1=-1
Now for each answer choice solve the function for both 2 and -1 and the answer choice where
f(2)=f(-1) would be our answer. Lets say if you encounter 2 answer choice where f(2)=f(-1) then select another number for x and solve those answer choices again using the new value for x and 1-x.

$$f(2)=2^2\times(1-2)^2=4\times1=4$$
$$f(-1)=(-1)^2\times(1-(-1))^2=1\times\2^2=1\times\4=4$$
$$Hence f(2)=f(-1)$$
Senior Manager
Joined: 07 Jan 2008
Posts: 383

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14 Jun 2009, 10:48
Jozu wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

(1) f(x) = 1-x
(2) f(x) = $$1-x^2$$
(3) f(x) = $$x^2-(1-x)^2$$
(4) f(x) = $$x^2(1-x)^2$$
(5) f(x) = $$x/(1-x)$$

[Reveal] Spoiler:
D

My way to solve such a question is to try all of 5 choices.
Clearly 4 is correct.

--== Message from GMAT Club Team ==--

This is not a quality discussion. It has been retired.

If you would like to discuss this question please re-post it in the respective forum. Thank you!

To review the GMAT Club's Forums Posting Guidelines, please follow these links: Quantitative | Verbal Please note - we may remove posts that do not follow our posting guidelines. Thank you.
Re: f(x)   [#permalink] 14 Jun 2009, 10:48
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