GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 18 Sep 2018, 16:54

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

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

Author Message
Intern
Joined: 18 Feb 2005
Posts: 46
For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

### Show Tags

Updated on: 25 Sep 2014, 23:48
2
9
00:00

Difficulty:

55% (hard)

Question Stats:

65% (01:27) correct 35% (01:32) wrong based on 313 sessions

### HideShow timer Statistics

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

Originally posted by njss750 on 20 Nov 2005, 06:33.
Last edited by Bunuel on 25 Sep 2014, 23:48, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
Director
Joined: 14 Sep 2005
Posts: 959
Location: South Korea
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

### Show Tags

20 Nov 2005, 07:42
Should be D. F(x)=(x^2) (1-x^2)

D. is the only function which is the greatest at 1/2.
_________________

Auge um Auge, Zahn um Zahn !

Intern
Joined: 01 Jun 2005
Posts: 40
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

### Show Tags

20 Nov 2005, 11:12
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
Manager
Joined: 30 Aug 2005
Posts: 181
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

### Show Tags

25 Nov 2005, 08:09
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1835
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

### Show Tags

25 Sep 2014, 23:06
_________________

Kindly press "+1 Kudos" to appreciate

Math Expert
Joined: 02 Sep 2009
Posts: 49206
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

### Show Tags

25 Sep 2014, 23:49
5
1
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.

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

OPEN DISCUSSION OF THIS QUESTION IS HERE: for-which-of-the-following-functions-f-is-f-x-f-1-x-for-85751.html
_________________
Non-Human User
Joined: 09 Sep 2013
Posts: 8063
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

### Show Tags

21 Jul 2018, 07:18
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: For which of the following functions f is f(x) = f(1-x) for all x? &nbs [#permalink] 21 Jul 2018, 07:18
Display posts from previous: Sort by

# Events & Promotions

 Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.