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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]

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

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New post 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 all x?  [#permalink]

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New post 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 all x?  [#permalink]

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New post 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 all x?  [#permalink]

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New post 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 all x?  [#permalink]

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New post 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 all x?  [#permalink]

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

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New post 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!

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

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New post 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.

D is the right answer.

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

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New post 27 Mar 2014, 10:06
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Thank you Bunuel.

I didn't understand till the time i read about the option of picking numbers.

Even then it took a while for me to understand that i have to check both the values (-1 and 2) and check if i get the same answer.

I think I am clear about this concept now.

Please do let me know if there are more similar questions besides the ones you mentioned.

Thank you

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

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

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New post 12 Jun 2016, 22:33
Given: f(x)= f(1-x)
This means if we replace x by 1 - x in the function, still the result should be same.

Checking each option:

A f(x)=1-x
f(1 - x) = 1 - (1-x) = x. Not equal to f(x)
INCORRECT

B f(x)=1-x^2
f(1-x) = 1- (1-x)^2 = 1 - (1 +x^2 - 2x). Not equal to f(x)
INCORRECT

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 to f(x)
INCORRECT

D f(x)=x^2(1-x)^2
f(1-x) = (1-x)^2(1 - 1 + x)^2 = (1-x)^2*x^2.
This is equal to f(x)
CORRECT

E f(x)= x/(1-x)
f(1-x) = 1-x/1-x + x = (1-x)/x. Not equal to f(x)
INCORRECT

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

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New post 30 Mar 2017, 16:24
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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)


Since we are not given any restrictions on the value of x, let’s let x = 1. Thus, we are determining for which of the following functions is f(1) = f(1-1), i.e., f(1) = f(0). Next, we can test each answer choice using our value x = 1.

A. f(x) = 1 - x

f(1) = 1 - 1 = 0

f(0) = 1 - 0 = 1

Since 0 does not equal 1, A is not correct.

B. f(x) = 1 - x^2

f(1) = 1 - 1^2 = 1 - 1 = 0

f(0) = 1 - 0^2 = 1 - 0 = 1

Since 0 does not equal 1, B is not correct.

C. f(x) = x^2 - (1 - x)^2

f(1) = 1^2 - (1 - 1)^2 = 1 - 0 = 1

f(0) = 0^2 - (1 - 0)^2 = 0 - 1 = -1

Since 1 does not equal -1, C is not correct.

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

f(1) = 1^2*(1 - 1)^2 = 1(0)= 0

f(0) = 0^2*(1 - 0)^2 = 0(2) = 0

Since 0 equals 0, D is correct.

Alternate Solution:

Let’s test each answer choice using x and 1 - x.

A. f(x) = 1 - x

f(x) = 1 - x

f(1 - x) = 1 - (1 - x) = x

Since 1 - x does not equal x, A is not correct.

B. f(x) = 1 - x^2

f(x) = 1 - x^2

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

Since 1 - x^2 does not equal 2x - x^2, B is not correct.

C. f(x) = x^2 - (1 - x)^2

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

Since 2x - 1 does not equal 1 - 2x, C is not correct.

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

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

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

Since x^2*(1 - x)^2 equals (1 - x)^2*x^2, D is correct.

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

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New post 18 Jul 2017, 17:37
Most people seem to suck at explaining these function related questions. Here is a good video explaining some function skills that helps on other questions. https://www.youtube.com/watch?v=T6-Zdr5w_bE

The key to these questions is understanding that f(x) is the function. Meaning that f(3) would mean everytime you see an x you sub in a 3. Or in this case we sub in an X-3. So for instance the first question is F(x) = 1-x............. Imagine the X is a blank or a blank parentheses waiting to be filled in by a number. so F(3) = 1-3 or F(x) = 1 - X or F (blank)= 1-(blank) or F(1-x) = 1-(1-x)
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For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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New post 24 Nov 2018, 17:24
Think of functions in terms of inputs and outputs. We want a function for which the input of x will lead to exactly the same result as the input of (1-x). Below is a video explanation

https://www.youtube.com/watch?v=4ZT8aZHWmdk

For a function question, it is almost always best to pick numbers, and to choose numbers that are small and manageable. Lets chose 1 for x. The quantity (1-x) would therefore equal 0.

If you put each of those inputs into the function in answer choice A you'll see very quicky that the two outputs are not equal to each other. You'll see very quickly that answer B is also incorrect. Answer C takes a bit more time. The correct answer is D.

The question if easiest to answer if you create separate columns for f(x) and f(1-x). Once again, a video explanation can be found here:
https://www.youtube.com/watch?v=4ZT8aZHWmdk
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Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

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New post 29 Dec 2018, 09:53
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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. \(f(x) = \frac{x}{(1 - x)}\)


Let's try plugging in an easy value for x. How about x = 0.
So, we can reword the question as, For which of the following functions is f(0)=f(1-0)
In other words, we're looking for a function such that f(0) = f(1)

A) f(x)=1-x
f(0)=1-0 = 1
f(1)=1-1 = 0
Since f(0) doesn't equal f(1), eliminate A

B) f(x) = 1 - x^2
f(0) = 1 - 0^2 = 1
f(1) = 1 - 1^2 = 0
Since f(0) doesn't equal f(1), eliminate B

C) f(x) = x^2 - (1-x)^2
f(0) = 0^2 - (1-0)^2 = -1
f(1) = 1^2 - (1-1)^2 = 1
Since f(0) doesn't equal f(1), eliminate C

D) f(x) = x^2(1-x)^2
f(0) = 0^2(1-0)^2 = 0
f(1) = 1^2(1-1)^2 = 0
Since f(0) equals f(1), keep D for now

E) f(x) = x/(1-x)
f(0) = 0/(1-0) = 0
f(1) = 1/(1-1) = undefined
Since f(0) doesn't equal f(1), eliminate E

Since only D satisfies the condition that f(x)=f(1-x) when x=0, the correct answer is D

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