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

 It is currently 16 Oct 2019, 12:41

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
TAGS:

Hide Tags

Manager
Joined: 05 Oct 2008
Posts: 217
For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

25 Oct 2009, 02:34
22
177
00:00

Difficulty:

45% (medium)

Question Stats:

68% (02:03) correct 32% (02:04) wrong based on 2791 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. $$f(x) = \frac{x}{(1 - x)}$$
Math Expert
Joined: 02 Sep 2009
Posts: 58381
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

28 Oct 2009, 16:17
140
141
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.
_________________
SVP
Status: Three Down.
Joined: 09 Jun 2010
Posts: 1824
Concentration: General Management, Nonprofit
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

03 Jul 2010, 23:15
13
5
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

$$f(x) = \frac{x}{1-x}$$
$$f(1-x) = \frac{1-x}{1-(1-x)}=\frac{1-x}{x}$$
General Discussion
Kaplan GMAT Instructor
Joined: 21 Jun 2010
Posts: 67
Location: Toronto
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

04 Jul 2010, 22:33
4
Bunuel:

sometimes, I almost die from the awesomeness of your posts.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9705
Location: Pune, India
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

29 Dec 2010, 21:38
22
4
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.
_________________
Karishma
Veritas Prep GMAT Instructor

Manager
Joined: 12 Oct 2011
Posts: 154
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

23 Jan 2012, 00:40
3
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.
_________________
Consider KUDOS if you feel the effort's worth it
Math Expert
Joined: 02 Sep 2009
Posts: 58381
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

12 Oct 2012, 03:41
1
1
Similar question to practice: http://gmatclub.com/forum/functions-pro ... ml#p717196
_________________
Senior Manager
Joined: 13 Aug 2012
Posts: 401
Concentration: Marketing, Finance
GPA: 3.23
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

13 Dec 2012, 00:40
1
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!

_________________
Impossible is nothing to God.
Manager
Joined: 12 Mar 2012
Posts: 79
Location: India
Concentration: Technology, Strategy
GMAT 1: 710 Q49 V36
GPA: 3.2
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

23 Mar 2013, 22:01
5
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.
Intern
Joined: 23 Jan 2014
Posts: 2
Location: India
WE: Consulting (Consulting)
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

27 Mar 2014, 10:06
1
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.

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

Thank you

Rajat
Math Expert
Joined: 02 Sep 2009
Posts: 58381
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

27 Mar 2014, 10:47
rajatsp wrote:
Please do let me know if there are more similar questions besides the ones you mentioned.

Thank you

Rajat

Here are several more:
for-which-of-the-following-functions-does-f-x-f-2-x-155813.html
for-which-of-the-following-does-f-a-f-b-f-a-b-164979.html
for-which-of-the-following-functions-f-is-f-x-f-1-x-for-85751.html

Hope this helps.
_________________
SVP
Joined: 06 Nov 2014
Posts: 1873
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

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
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8069
Location: United States (CA)
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

30 Mar 2017, 16:24
1
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.

_________________

Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager
Joined: 31 Dec 2016
Posts: 68
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

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)
Attachments

Function question 6.png [ 417.34 KiB | Viewed 11580 times ]

Intern
Joined: 20 Aug 2018
Posts: 25
For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

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

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:
_________________
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 4006
Re: For which of the following functions f is f(x) = f(1-x) for all x?  [#permalink]

Show Tags

29 Dec 2018, 09:53
1
Top Contributor
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

Cheers,
Brent
_________________
Test confidently with gmatprepnow.com
Non-Human User
Joined: 09 Sep 2013
Posts: 13210
Re: For which of the following functions f is f(x) =  [#permalink]

Show Tags

09 Oct 2019, 04:26
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) =   [#permalink] 09 Oct 2019, 04:26
Display posts from previous: Sort by