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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]

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25 Oct 2009, 02:34
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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)
[Reveal] Spoiler: OA
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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28 Oct 2009, 04:18
Can someone please solve this. I know what the answer is. I am looking for an explanation on how to get to the answer.

Bunuel?

Thanks.
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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28 Oct 2009, 16:17
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study 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)

I've already seen quite a few GMAT questions of this type. They are quite easy to solve, since you understand the concept.

$$f(x)="some \ expression \ with \ variable \ x"$$, 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.

Similar questions:
for-which-of-the-following-functions-is-f-a-b-f-a-f-b-for-93184.html
for-which-of-the-following-functions-is-f-a-b-f-b-f-a-124491.html
let-the-function-g-a-b-f-a-f-b-143311.html

Hope it helps.
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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29 Oct 2009, 04:29
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Helps very much

Thanks to Lagomez and Bunuel..
+1
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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03 Jul 2010, 23:15
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Okay, so you need to check if f(x) = f(1-x)

Let's do this one by one.

Case A: - Wrong
$$f(x) = 1-x$$
$$f(1-x) = 1-(1-x) = x$$

So we clearly see that f(x) is not f(1-x)

Case B: Wrong

$$f(x) = 1-x-x^2$$
$$f(1-x) = 1-(1-x)-(1-x)^2 = x - 1 - x^2 + 2x = 3x-1-x^2$$

Case C: Wrong

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

Case D: Right

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

Let's check E just to be sure.

Case E: Wrong

$$f(x) = \frac{x}{1-x}$$
$$f(1-x) = \frac{1-x}{1-(1-x)}=\frac{1-x}{x}$$
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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04 Jul 2010, 16:07
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Bunuel wrote:
Merging similar topics.

Your contributions make the forum very resourceful.
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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04 Jul 2010, 22:33
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Bunuel:

sometimes, I almost die from the awesomeness of your posts.
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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29 Dec 2010, 21:38
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metallicafan wrote:
Is there a faster way to solve this question rather than replacing each "x" by (1-x)? Thanks!

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. $$f(x) = x / (1-x)$$

Tip: Try to first focus on the options where terms are added/multiplied rather than subtracted/divided. They are more symmetrical and a substitution may not change the expression. I will intuitively check D first since it involves multiplication of the terms.
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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16 Jan 2011, 10:48
Thanks guys, great help!
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03 May 2011, 11:01
f(x) = f(1-x)

1) 1-x = 1-(1-x) = 1-1+x = x ==> Not possible
2) 1-x^2 = 1-(1-x)^2 = 2x-x^2 ==> Not possible
3) x^2 - (1-x)^2 = (1-x) ^ 2 - (1-(1-x))^2 = (1-x)^2-x^2 ==> Not possible (sign difference)
4) x^2 (1-x)^2 = (1-x)^2 (1-(1-x))^2 = (1-x)^2 x^2 ==>Works
5) x / 1-x = 1-x/ 1-(1-x) = 1-x/x ==> Not possible

Hence the correct answer is D.
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03 May 2011, 20:37
for D,

substituting for x, we have (1-x)^2 * x^2 = f(x)

Hence the OA.
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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23 Jan 2012, 00:40
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This one is quite simple. We have to substitute (1 - x) for x in each of the options and see which option yields the same result for both f(x) and f(1 - x).

However, careful observation of the options will easily tell you that only D will fulfill this condition because in D, the x now becomes (1 - x) and (1 - x) now becomes x. The overall outcome will remain the same. You don't even need to expand or do any calculations whatsoever.
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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23 Jan 2012, 01:40
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Yes indeed simple after explanations, but was difficult to understand. Many thanks
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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12 Oct 2012, 03:41
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Expert's post
IanSolo 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)=$$\frac{x}{1-x}$$

I find this question in gmat prep software, I have tried to solve it with picking number ( I have sostitute x with 1 and -1) but for all the answer the result was different from 1 and -1.
How I can solve quickly this question?
What is the level of this question?

Similar question to practice: functions-problem-need-help-93184.html#p717196
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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13 Dec 2012, 00:40
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A.
$$1-x ? 1-(1-x)$$
$$1-x ? 1-1+x$$
$$1-x ? x$$NOT EQUAL!

B.
$$1-x^2 ? 1-(1-x)^2$$
$$1-x^2 ? 2x + x^2$$ NOT EQUAL!

C.
$$x^2 - (1-x)^2 ? (1-x)^2 - (1-(1-x))^2$$
$$x^2 - (1-x)^2 ? (1-x)^2 - x^2$$ NOT EQUAL!

D.
$$x^2(1-x)^2 ? (1-x)^2(1-(1-x))^2$$
$$x^2(1-x)^2 ? (1-x)^2(x)^2$$ EQUAL!

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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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27 Feb 2013, 03:42
Karishma....that is a useful tip....does it work in majority of the problems and thus safe to follow every time any such problem crops up???
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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23 Mar 2013, 22:01
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You need to check for every option.
Either you can replace x in every option with 1-x or you can take x = 1 and then check for which option f(1) = f(1-1) = f(0).
In A f(1) = 1-1 = 0. f(0) = 1-0 = 1.
In B f(1) = 1 - 1^2 = 0. f(0) = 1 - 0 = 1
In C f(1) = 1^2 - (1 - 1 )^2 = 1. f(0) = 0 - (1-0)^2 = 0 - 1 = -1
in D f(1) = 1(1-1)^2 = 0. f(0) = 0.(1-0)^2 = 0. ( Right answer )
And for option E you can take x = 2 because if you take x = 1 in denominator the denominator becomes zero, which makes the f(x) undefined.
f(2) = 2/(1 - 2) = -2. f(1-2) = f(-1) = -1/(1-(-1)) = -1/2.

Please give a kudo if you like my explanation.
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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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04 Aug 2013, 08:25
smrutipattnaik wrote:
For which of the following fns 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)

Option A
f(x)=1-x
f(1-x) = 1- (1-x) =x
Not Equal

Option B
f(x)=1-x^2
f(1-x) = 1- (1-x)^2 =1- (1-2x +x^2) = 2x -x^2
Not Equal

Option C
f(x)=x^2-(1-x)^2
f(1-x) = (1-x)^2- (1-1+x)^2 =(1-x)^2- (x)^2
Not Equal

Option D
f(x)=x^2(1-x)^2
f(1-x) = (1-x)^2 (1-1+x)^2 =(1-x)^2 (x)^2
Equal

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Re: For which of the following functions f is f(x)=f(1-x) for [#permalink]

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25 Dec 2013, 18:31
Replacing x=4 and 1-x = -3 I get 16(9) - why is that wrong
4^2 (1-4)^2 = 16 (9), not at all f(4)
Re: For which of the following functions f is f(x)=f(1-x) for   [#permalink] 25 Dec 2013, 18:31

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