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

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For which of the following functions f is f(x) = f(1 - x) [#permalink] New post 21 Aug 2005, 18:27
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)
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Re: Function [#permalink] New post 22 Aug 2005, 06:11
Folaa3 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)



4. Only it has 'x' and '1-x' listed as symetrically
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Re: Function [#permalink] New post 23 Aug 2005, 05:47
Yes, that is a very good way to view it. For people who want to make absolutely sure, or for more complicated problems, one would have to actually plug it in and try it out.

1. f(x) = 1-x
f(1-x)=1-(1-x)=x
No

2. f(x) = 1 - x^2
f(1-x)=1-(1-x)^2
No

3. f(x) = x^2 - (1 - x) ^2
f(1-x)=(1-x)^2-x^2=-f(x)
No

4. f(x) = x^2 ( 1 - x) ^2
f(1-x)=(1-x)^2x^2=f(x)
Yes

5. f(x) = x / (1-x)
f(1-x)=(1-x)/x
No
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 [#permalink] New post 23 Aug 2005, 06:52
Just plug in the values! :) If the algebra looks bad, use some non-trivial numbers and see. For instance use x = 3, 1-x=-2
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 [#permalink] New post 23 Aug 2005, 06:58
Or better yet use x=0 and 1-x=1. However to use number plugins you need to be very very careful. If it turns out that they produce different values then you are ok, as only one counter example would be sufficient to say No. If it turns out that they do produce the same value, however, you can't simply say yes, for there may be other numbers that would make the two functions produce different values.
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Re: Function [#permalink] New post 23 Aug 2005, 12:15
For these kind of questions, the fastest way is to substitute 1-x in place of x in the f(x) ...see below....OA shud be "4".Plugging shud be the last measure as it can lead to erroneous conclusions and it takes time.

Folaa3 wrote:
For which of the following functions f is f(x) = f(1 - x) for all x?

1. f(x) = 1-x ---> f(1-x) = 1-(1-x) = x <> 1-x ....out
2. f(x) = 1 - x^2 ----> f(1-x) = 1-(1-x)^2 = 2x - x^2 <> 1-x^2 ...out

3. f(x) = x^2 - (1 - x) ^2 ----> f(1-x) = (1-x)^2 - x^2 ....no always true
4. f(x) = x^2 ( 1 - x) ^2 ---> f(1-x) = (1-x)^2 x^2 = f(x).....right

5. f(x) = x / (1-x) ----> f(1-x) = (1-x)/x.....<> f(x) .....not true
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 [#permalink] New post 23 Aug 2005, 14:28
4) SUbstituting (x-1) in place of x gives the same value....
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  [#permalink] 23 Aug 2005, 14:28
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For which of the following functions f is f(x) = f(1 - x)

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