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

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For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 28 Jun 2011, 08:12
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For which of the following functions f is f(x) = f(1-x) for all x?

a) f(x) = 1-x

b) f(x) = 1-x^2

c) f(x) = x^2 - (1-x)^2

d) f(x) = x^2 * (1-x)^2

e) f(x) = x/(1-x)

OPEN DISCUSSION OF THIS QUESTION IS HERE: for-which-of-the-following-functions-f-is-f-x-f-1-x-for-85751.html
[Reveal] Spoiler: OA

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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 28 Jun 2011, 08:17
It's D, a quick inspection reveals the symmetry of the expressions.

f(x) = x^2 * (1-x)^2

Substitute x with (1-x) :

f(1-x) = (1-x)^2 (1-1+x)^2 = (1-x)^2 * x^2 = f(x)
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 28 Jun 2011, 08:23
subhashghosh wrote:
It's D, a quick inspection reveals the symmetry of the expressions.

f(x) = x^2 * (1-x)^2

Substitute x with (1-x) :

f(1-x) = (1-x)^2 (1-1+x)^2 = (1-x)^2 * x^2 = f(x)


please make me understand the "quick inspection process."
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 28 Jun 2011, 08:27
I meant a quick scan of choices in general :)
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 28 Jun 2011, 08:29
Baten80 wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

a) f(x) = 1-x

b) f(x) = 1-x^2

c) f(x) = x^2 - (1-x)^2

d) f(x) = x^2 * (1-x)^2

e) f(x) = x/(1-x)


u need to try each option.. :(
a. 1-x = 1-(1-x)= X ............ No
b 1-x^2 = same as above ........ No
3. X^2-(1-X)^2 = (1-x)^2- (1-1+X)^2
= (1-x)^2- X^2............. No
4. x^2*(1-x)^2= (1-x)^2(1-(1-X))^2
= (1-x)^2*x^2............... yes
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 28 Jun 2011, 08:36
sudhir18n wrote:
Baten80 wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

a) f(x) = 1-x

b) f(x) = 1-x^2

c) f(x) = x^2 - (1-x)^2

d) f(x) = x^2 * (1-x)^2

e) f(x) = x/(1-x)


u need to try each option.. :(
a. 1-x = 1-(1-x)= X ............ No
b 1-x^2 = same as above ........ No
3. X^2-(1-X)^2 = (1-x)^2- (1-1+X)^2
= (1-x)^2- X^2............. No
4. x^2*(1-x)^2= (1-x)^2(1-(1-X))^2
= (1-x)^2*x^2............... yes


Basically I did it at the test.
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 28 Jun 2011, 08:40
We don't need to try every option, e.g., A and E are not possible is clear at the first glance.
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 28 Jun 2011, 09:09
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Baten80 wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

a) f(x) = 1-x

b) f(x) = 1-x^2

c) f(x) = x^2 - (1-x)^2

d) f(x) = x^2 * (1-x)^2

e) f(x) = x/(1-x)


For

x=1;
1-x=0

a) f(1)=1-1=0; f(0)=1-0=1
b) f(1)=1-1=0; f(0)=1-0=1
c) f(1)=1-(1-1)^2=1-0=1; f(0)=0-(1-0)^2=-1
d) f(1)=1*(1-1)^2=0; f(0)=0---- Correct. f(1)=f(0)
e) f(1)=1/0=undefined; f(0)=0

Ans: "D"
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 28 Jun 2011, 09:43
I also got this same question on my yesterday gmat prep practice test1 and I tried the same way as Fluke did and ended up with answer shoice D. :)

sometime we have to put some number in the answer choice and see if that matches with the question asked ...

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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 29 Jun 2011, 17:19
substituting x with 1-x and verifying the f(x) and f(1-x) in the answer choices, we can see that Answer is D.
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 29 Jun 2011, 17:37
I arrived at D but over the 2 minute benchmark. Seems like I need to work on finding ways to eliminate choices off the bat as I tested each option for this question.
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 07 Aug 2011, 05:02
fluke wrote:
Baten80 wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

a) f(x) = 1-x

b) f(x) = 1-x^2

c) f(x) = x^2 - (1-x)^2

d) f(x) = x^2 * (1-x)^2

e) f(x) = x/(1-x)


For

x=1;
1-x=0

a) f(1)=1-1=0; f(0)=1-0=1
b) f(1)=1-1=0; f(0)=1-0=1
c) f(1)=1-(1-1)^2=1-0=1; f(0)=0-(1-0)^2=-1
d) f(1)=1*(1-1)^2=0; f(0)=0---- Correct. f(1)=f(0)
e) f(1)=1/0=undefined; f(0)=0

Ans: "D"


Thanks for directing me here fluke..need your help with point c and d..

we took x=1 and 1-x=0..if so then c should be
f(x) = x^2 - (1-x)^2 -------- Original Statement
1 = 1 - (0)^2
1=1

and d should be
f(x) = x^2 * (1-x)^2 -------- Original Statement
1 = 1 * (0)^2
1=0

I am sure there is a miss in my understanding for which i need your help.
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 07 Aug 2011, 05:30
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DeeptiM wrote:
fluke wrote:
Baten80 wrote:
For which of the following functions f is f(x) = f(1-x) for all x?

a) f(x) = 1-x

b) f(x) = 1-x^2

c) f(x) = x^2 - (1-x)^2

d) f(x) = x^2 * (1-x)^2

e) f(x) = x/(1-x)


For

x=1;
1-x=0

a) f(1)=1-1=0; f(0)=1-0=1
b) f(1)=1-1=0; f(0)=1-0=1
c) f(1)=1-(1-1)^2=1-0=1; f(0)=0-(1-0)^2=-1
d) f(1)=1*(1-1)^2=0; f(0)=0---- Correct. f(1)=f(0)
e) f(1)=1/0=undefined; f(0)=0

Ans: "D"


Thanks for directing me here fluke..need your help with point c and d..

we took x=1 and 1-x=0..if so then c should be
f(x) = x^2 - (1-x)^2 -------- Original Statement
1 = 1 - (0)^2 {:Note: Here LHS should be f(1) not 1. You need to consider both case one by one}
1=1

and d should be
f(x) = x^2 * (1-x)^2 -------- Original Statement
1 = 1 * (0)^2
1=0

I am sure there is a miss in my understanding for which i need your help.


If there is a function;
f(x)=x-1
f(1)=1-1=0
f(2)=2-1=0
f(z)=z-1
f(1+x)=1+x-1=x

You see how we replace x with whatever is in the bracket of f().

Likewise;
f(x) = x^2 - (1-x)^2
f(1)=(1)^2-(1-1)^2=1-0=1 {:Note: Replace x with 1}
f(0)=0^2-(1-0)^2=0-1=-1 {:Note: Replace x with 0}

You see:
f(1) NOT EQUAL f(0). Thus, it is incorrect.

We can't do partial replacement.

Or algebraically:

C:
f(x) = x^2 - (1-x)^2= x^2-(1+x^2-2x)=x^2-1-x^2+2x=2x-1
f(1-x)=(1-x)^2 - (1-(1-x))^2=1+x^2-2x-x^2=1-2x {:Note: I replaced x with 1-x everywhere}

Try the same with D yourself.
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 07 Aug 2011, 06:06
Awsum..i get it now..

Thanks Fluke..for the great explanation and above all your patience!!
Kudos to you!
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 27 Feb 2013, 03:06
As a General Strategy is it good to test 0 and 1 for these questions?
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Re: For which of the following functions f is f(x) = f(1-x) for [#permalink] New post 27 Feb 2013, 03:25
fozzzy wrote:
As a General Strategy is it good to test 0 and 1 for these questions?


Check this post: for-which-of-the-following-functions-f-is-f-x-f-1-x-for-85751.html#p644387 (plug in numbers method)
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Re: For which of the following functions f is f(x) = f(1-x) for   [#permalink] 27 Feb 2013, 03:25
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