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# M11-24

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Math Expert
Joined: 02 Sep 2009
Posts: 51035

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15 Sep 2014, 23:45
3
13
00:00

Difficulty:

75% (hard)

Question Stats:

55% (01:21) correct 45% (01:27) wrong based on 144 sessions

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Function $$F(x)$$ is defined as follows:

if $$x$$ is positive or 0 then $$F(x) = 1 - x$$;

if $$x$$ is negative then $$F(x) = x - 1$$.

Which of the following is true about $$F(x)$$?

I. $$F(|x|) = |F(x)|$$

II. $$F(x) = F(-x)$$

III. $$|F(x + 1)| = |x|$$

A. I only
B. II only
C. III only
D. I and III only
E. not I, II, or III

_________________
Math Expert
Joined: 02 Sep 2009
Posts: 51035

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15 Sep 2014, 23:45
1
3
Official Solution:

Function $$F(x)$$ is defined as follows:

if $$x$$ is positive or 0 then $$F(x) = 1 - x$$;

if $$x$$ is negative then $$F(x) = x - 1$$.

Which of the following is true about $$F(x)$$?

I. $$F(|x|) = |F(x)|$$

II. $$F(x) = F(-x)$$

III. $$|F(x + 1)| = |x|$$

A. I only
B. II only
C. III only
D. I and III only
E. not I, II, or III

I is not necessarily true. Consider $$x = -1$$. $$F(|x|) = 0$$ does not equal $$|F(x)| = 2$$.

II is not necessarily true. Consider $$x = 1$$. $$F(x) = 0$$ does not equal $$F(-x) = -2$$.

III is always true:

if $$x + 1$$ is non-negative then $$|F(x + 1)| = |1 - (x + 1)| = |-x| = |x|$$;

if $$x + 1$$ is negative then $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

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Joined: 12 Sep 2010
Posts: 233
Concentration: Healthcare, General Management

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23 Oct 2014, 12:27
1
I think the question should be change to "Which of the following MUST be True about f(x)?" to make it more clear.
Math Expert
Joined: 02 Sep 2009
Posts: 51035

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24 Oct 2014, 02:18
Samwong wrote:
I think the question should be change to "Which of the following MUST be True about f(x)?" to make it more clear.

Is true and must be true are the same and used interchangeably. Check the following problems from Official Guides:
in-the-figure-above-if-f-is-a-point-on-the-line-that-bisect-165054.html
m-is-the-sum-of-the-reciprocals-of-the-consecutive-integers-143703.html
if-a-0-3-which-of-the-following-is-true-143992.html
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Joined: 12 Sep 2010
Posts: 233
Concentration: Healthcare, General Management

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24 Oct 2014, 11:41
Cool, I did not know that. Thank Bunuel for the reply.
Manager
Joined: 08 Feb 2014
Posts: 204
Location: United States
Concentration: Finance
GMAT 1: 650 Q39 V41
WE: Analyst (Commercial Banking)

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12 Nov 2014, 20:39
Any quick approaches here? Plug and chug is taking me a bit too long...

Thanks
Retired Moderator
Status: The best is yet to come.....
Joined: 10 Mar 2013
Posts: 502

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17 Jan 2015, 01:17
Bunuel wrote:
Official Solution:

Function $$F(x)$$ is defined as follows:

if $$x$$ is positive or 0 then $$F(x) = 1 - x$$;

if $$x$$ is negative then $$F(x) = x - 1$$.

Which of the following is true about $$F(x)$$?

I. $$F(|x|) = |F(x)|$$

II. $$F(x) = F(-x)$$

III. $$|F(x + 1)| = |x|$$

A. I only
B. II only
C. III only
D. I and III only
E. not I, II, or III

I is not necessarily true. Consider $$x = -1$$. $$F(|x|) = 0$$ does not equal $$|F(x)| = 2$$.

II is not necessarily true. Consider $$x = 1$$. $$F(x) = 0$$ does not equal $$F(-x) = -2$$.

III is always true:

if $$x + 1$$ is non-negative then $$|F(x + 1)| = |1 - (x + 1)| = |-x| = |x|$$;

if $$x + 1$$ is negative then $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Anyone to clarify the following? I don't understand this.

if x + 1 is non-negative then |F(x + 1)| = |1 - (x + 1)| = |-x| = |x|;

if x + 1 is negative then |F(x + 1)| = |(x + 1) - 1| = |x|

_________________

Hasan Mahmud

Math Expert
Joined: 02 Sep 2009
Posts: 51035

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17 Jan 2015, 16:00
Mahmud6 wrote:
Bunuel wrote:
Official Solution:

Function $$F(x)$$ is defined as follows:

if $$x$$ is positive or 0 then $$F(x) = 1 - x$$;

if $$x$$ is negative then $$F(x) = x - 1$$.

Which of the following is true about $$F(x)$$?

I. $$F(|x|) = |F(x)|$$

II. $$F(x) = F(-x)$$

III. $$|F(x + 1)| = |x|$$

A. I only
B. II only
C. III only
D. I and III only
E. not I, II, or III

I is not necessarily true. Consider $$x = -1$$. $$F(|x|) = 0$$ does not equal $$|F(x)| = 2$$.

II is not necessarily true. Consider $$x = 1$$. $$F(x) = 0$$ does not equal $$F(-x) = -2$$.

III is always true:

if $$x + 1$$ is non-negative then $$|F(x + 1)| = |1 - (x + 1)| = |-x| = |x|$$;

if $$x + 1$$ is negative then $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Anyone to clarify the following? I don't understand this.

if x + 1 is non-negative then |F(x + 1)| = |1 - (x + 1)| = |-x| = |x|;

if x + 1 is negative then |F(x + 1)| = |(x + 1) - 1| = |x|

We are told that:
if $$x$$ is positive or 0 then $$F(x) = 1 - x$$;
if $$x$$ is negative then $$F(x) = x - 1$$.

So,
if x + 1 is non-negative then |F(x + 1)| = |1 - (x + 1)| = |-x| = |x|;
if x + 1 is negative then |F(x + 1)| = |(x + 1) - 1| = |x|.
_________________
Intern
Joined: 16 May 2016
Posts: 3

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11 Jul 2016, 06:16
Bunuel wrote:
Official Solution:

Function $$F(x)$$ is defined as follows:

if $$x$$ is positive or 0 then $$F(x) = 1 - x$$;

if $$x$$ is negative then $$F(x) = x - 1$$.

Which of the following is true about $$F(x)$$?

I. $$F(|x|) = |F(x)|$$

II. $$F(x) = F(-x)$$

III. $$|F(x + 1)| = |x|$$

A. I only
B. II only
C. III only
D. I and III only
E. not I, II, or III

I is not necessarily true. Consider $$x = -1$$. $$F(|x|) = 0$$ does not equal $$|F(x)| = 2$$.

II is not necessarily true. Consider $$x = 1$$. $$F(x) = 0$$ does not equal $$F(-x) = -2$$.

III is always true:

if $$x + 1$$ is non-negative then $$|F(x + 1)| = |1 - (x + 1)| = |-x| = |x|$$;

if $$x + 1$$ is negative then $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Hi Bunuel, the highlighted answer in red shouldn't be "2"?
As X=1, we should consider the positive branch of the function, so f(-x)=1-(-x) would result in f(-1)=1-(-1)=2.
Am I wrong?

Thanks!
Math Expert
Joined: 02 Sep 2009
Posts: 51035

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11 Jul 2016, 06:21
RodrigoMi wrote:
Bunuel wrote:
Official Solution:

Function $$F(x)$$ is defined as follows:

if $$x$$ is positive or 0 then $$F(x) = 1 - x$$;

if $$x$$ is negative then $$F(x) = x - 1$$.

Which of the following is true about $$F(x)$$?

I. $$F(|x|) = |F(x)|$$

II. $$F(x) = F(-x)$$

III. $$|F(x + 1)| = |x|$$

A. I only
B. II only
C. III only
D. I and III only
E. not I, II, or III

I is not necessarily true. Consider $$x = -1$$. $$F(|x|) = 0$$ does not equal $$|F(x)| = 2$$.

II is not necessarily true. Consider $$x = 1$$. $$F(x) = 0$$ does not equal $$F(-x) = -2$$.

III is always true:

if $$x + 1$$ is non-negative then $$|F(x + 1)| = |1 - (x + 1)| = |-x| = |x|$$;

if $$x + 1$$ is negative then $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Hi Bunuel, the highlighted answer in red shouldn't be "2"?
As X=1, we should consider the positive branch of the function, so f(-x)=1-(-x) would result in f(-1)=1-(-1)=2.
Am I wrong?

Thanks!

If $$x = 1$$. $$F(-x) =F(-1)$$. We know that if $$x$$ is negative then $$F(x) = x - 1$$, thus $$F(-x) =F(-1)=-1-1=-2$$.
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Manager
Joined: 23 Jun 2009
Posts: 181
Location: Brazil
GMAT 1: 470 Q30 V20
GMAT 2: 620 Q42 V33

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03 Aug 2016, 15:18
birsencal wrote:
Please explain this in more detail

Hope it helps
>> !!!

You do not have the required permissions to view the files attached to this post.

Intern
Joined: 01 Nov 2017
Posts: 29

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02 Jul 2018, 20:41
Bunuel wrote:
Official Solution:

III is always true:

if $$x + 1$$ is non-negative then $$|F(x + 1)| = |1 - (x + 1)| = |-x| = |x|$$;

if $$x + 1$$ is negative then $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

I'm not following this at all. Can someone explain further? How do you get the 1 in 1-(x+1) for instance?
Math Expert
Joined: 02 Sep 2009
Posts: 51035

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02 Jul 2018, 20:54
surfingpirate wrote:
Bunuel wrote:
Official Solution:

III is always true:

if $$x + 1$$ is non-negative then $$|F(x + 1)| = |1 - (x + 1)| = |-x| = |x|$$;

if $$x + 1$$ is negative then $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

I'm not following this at all. Can someone explain further? How do you get the 1 in 1-(x+1) for instance?

We are told that:
if $$x$$ is positive or 0 then $$F(x) = 1 - x$$;
if $$x$$ is negative then $$F(x) = x - 1$$.

So,
if x + 1 is non-negative then $$|F(x + 1)| = |1 - (x + 1)| = |-x| = |x|$$;
if x + 1 is negative then $$|F(x + 1)| = |(x + 1) - 1| = |x|$$.
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Intern
Joined: 11 Feb 2017
Posts: 3

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07 Jul 2018, 03:56
1
Hi Brunnel,

Plz guide me where am I going wrong

for Option I : F (|x|) will always be equal to 1-x as |x| will always be greater than or equal to 0.

For x<0 : |F(x)| = |x-1|= |-(1-x)|= 1-x
For x>=0 : |F(x)| = |1-x| = 1-x

Therefore F (|x|) = |F(x)|

I understand you want me to try and negate the given must be condition. Can it be done algebraically without substitution ?
Intern
Joined: 07 Apr 2018
Posts: 1

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15 Oct 2018, 10:10
shasankashekhar wrote:
Hi Brunnel,

Plz guide me where am I going wrong

for Option I : F (|x|) will always be equal to 1-x as |x| will always be greater than or equal to 0.

For x<0 : |F(x)| = |x-1|= |-(1-x)|= 1-x
For x>=0 : |F(x)| = |1-x| = 1-x

Therefore F (|x|) = |F(x)|

I understand you want me to try and negate the given must be condition. Can it be done algebraically without substitution ?

Hi @bunnuel : any reason why this approach is wrong, got stuck in the same solution
Re: M11-24 &nbs [#permalink] 15 Oct 2018, 10:10
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# M11-24

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