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

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For which of the following functions f is f(x) = f(1-x) for all x? [#permalink] New post 20 Nov 2005, 05:33
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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

Last edited by Bunuel on 25 Sep 2014, 22:48, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: For which of the following functions f is f(x) = f(1-x) for all x? [#permalink] New post 20 Nov 2005, 06:42
Should be D. F(x)=(x^2) (1-x^2)

D. is the only function which is the greatest at 1/2.
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Re: For which of the following functions f is f(x) = f(1-x) for all x? [#permalink] New post 20 Nov 2005, 10:12
njss750 wrote:
For which of the following functions f is f(x)= f(1-x) for all x
F(x)=1-x
F(x)=1-x^2
F(x)=x^2-(1-x)^2
F(x)=(x^2) (1-x^2)
F(x)-x/ 1-x


I remember seeing this problem in a GMATprep test. I think choice D should be F(x)=(x^2) (1-x)^2. If that is true then substituting (1-x) for will give us the same function back and the answer choice is D, where the function is multiplicative
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Re: For which of the following functions f is f(x) = f(1-x) for all x? [#permalink] New post 25 Nov 2005, 07:09
Agreed, none of the original answers made sense.
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Re: For which of the following functions f is f(x) = f(1-x) for all x? [#permalink] New post 25 Sep 2014, 22:06
Can someone please update the OA / answers?
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Re: For which of the following functions f is f(x) = f(1-x) for all x? [#permalink] New post 25 Sep 2014, 22:49
Expert's post
njss750 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)


f(x)="some \ expression \ with \ variable \ x", means that the value of f(x) can be found by calculating the expression for the particular x.

For example: if f(x)=3x+2, what is the value of f(3)? Just plug 3 for x, f(3)=3*3+2=11, so if the function is f(x)=3x+2, then f(3)=11.

There are some functions for which f(x)=f(-x). For example: if we define f(x) as f(x)=3*x^2+2, the value of f(x) will be always positive and will give the following values: for x=-5, f(x)=3*(-5)^2+2=77; for x=0, f(0)=3*0^2+2=2. Please note that f(x) in this case is equal to f(-x), meaning that for positive values of x you'll get the same values of f(x) as for the negative values of x.

So, basically in original question we are told to define the expression, for which f(x)=f(1-x), which means that plugging x and 1-x in the expression must give same result.

A. f(x)=1-x --> 1-x is the expression for f(x), we want to find whether the expression for f(1-x) would be the same: plug 1-x --> f(1-x)=1-(1-x)=x. As 1-x and x are different, so f(x) does not equal to f(1-x).

The same with the other options:

(A) f(x)=1-x, so f(1-x)=1-(1-x)=x --> 1-x and x: no match.

(B) f(x)=1-x^2, so f(1-x)=1-(1-x)^2=1-1+2x-x^2=2x-x^2 --> 1-x^2 and 2x-x^2: no match.

(C) f(x)=x^2-(1-x)^2=x^2-1+2x+x^2=2x-1, so f(1-x)=2(1-x)-1=1-2x --> 2x-1 and 1-2x: no match.

(D) f(x)=x^2*(1-x)^2, so f(1-x)=(1-x)^2*(1-1+x)^2=(1-x)^2*x^2 --> x^2*(1-x)^2 and (1-x)^2*x^2. Bingo! if f(x)=x^2*(1-x)^2 then f(1-x) also equals to x^2*(1-x)^2.

Still let's check (E)

(E) f(x)=\frac{x}{1-x} --> f(1-x)=\frac{1-x}{1-1+x}=\frac{1-x}{x}. \frac{x}{1-x} and \frac{1-x}{x}: no match.

But this problem can be solved by simple number picking: plug in numbers.

As stem says that "following functions f is f(x) = f (1-x) for all x", so it should work for all choices of x.

Now let x be 2 (note that: -1, 0, and 1 generally are not good choices for number picking), then 1-x=1-2=-1. So we should check whether f(2)=f(-1).

(A) f(2)=1-x=1-2=-1 and f(-1)=1-(-1)=2 --> -1\neq{2};

(B) f(2)=1-x^2=1-4=-3 and f(-1)=1-1=0 --> -3\neq{0};

(C) f(2)=x^2-(1-x)^2=x^2-1+2x+x^2=2x-1=2*2-1=3 and f(-1)=2*(-1)-1=-3 --> 3\neq{-3};

(D) f(2)=x^2*(1-x)^2=(-2)^2*(-1)^2=4 and f(-1)=(-1)^2*2^2=4 --> 4=4, correct;

(E) f(2)=\frac{x}{1-x}=\frac{2}{1-2}=-2 and f(-1)=\frac{-1}{1-(-1)}=-\frac{1}{2} --> -2\neq{-\frac{1}{2}}.

It might happen that for some choices of x other options may be "correct" as well. If this happens, just pick some other number for x and check again these "correct" options only.

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OPEN DISCUSSION OF THIS QUESTION IS HERE: for-which-of-the-following-functions-f-is-f-x-f-1-x-for-85751.html
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Re: For which of the following functions f is f(x) = f(1-x) for all x?   [#permalink] 25 Sep 2014, 22:49
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