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Intern
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Joined: 08 Jun 2009
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f(x) [#permalink]

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New post 14 Jun 2009, 00:15
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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
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Joined: 03 Aug 2006
Posts: 115

Kudos [?]: 305 [0], given: 3

Location: Next to Google
Schools: Haas School of Business
Re: f(x) [#permalink]

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New post 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.

For answer choice 4
\(f(2)=2^2\times(1-2)^2=4\times1=4\)
\(f(-1)=(-1)^2\times(1-(-1))^2=1\times\2^2=1\times\4=4\)
\(Hence f(2)=f(-1)\)

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Senior Manager
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Joined: 07 Jan 2008
Posts: 398

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Re: f(x) [#permalink]

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New post 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.

Kudos [?]: 292 [0], given: 0

Re: f(x)   [#permalink] 14 Jun 2009, 10:48
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