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Math Expert V
Joined: 02 Sep 2009
Posts: 55228
M11-24  [#permalink]

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3
13 00:00

Difficulty:   75% (hard)

Question Stats: 53% (01:22) correct 47% (01:30) wrong based on 153 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 V
Joined: 02 Sep 2009
Posts: 55228
Re M11-24  [#permalink]

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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 $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Answer: C
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Manager  B
Joined: 12 Sep 2010
Posts: 230
Concentration: Healthcare, General Management
Re: M11-24  [#permalink]

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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 V
Joined: 02 Sep 2009
Posts: 55228
Re: M11-24  [#permalink]

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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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Manager  B
Joined: 12 Sep 2010
Posts: 230
Concentration: Healthcare, General Management
Re: M11-24  [#permalink]

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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)
Re: M11-24  [#permalink]

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Any quick approaches here? Plug and chug is taking me a bit too long...

Thanks
Retired Moderator P
Status: The best is yet to come.....
Joined: 10 Mar 2013
Posts: 494
Re: M11-24  [#permalink]

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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 $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Answer: C

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 V
Joined: 02 Sep 2009
Posts: 55228
Re: M11-24  [#permalink]

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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 $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Answer: C

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  B
Joined: 16 May 2016
Posts: 3
Re: M11-24  [#permalink]

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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 $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Answer: C

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 V
Joined: 02 Sep 2009
Posts: 55228
Re: M11-24  [#permalink]

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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 $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Answer: C

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  B
Joined: 23 Jun 2009
Posts: 177
Location: Brazil
GMAT 1: 470 Q30 V20 GMAT 2: 620 Q42 V33 Re: M11-24  [#permalink]

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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  B
Joined: 01 Nov 2017
Posts: 30
Re: M11-24  [#permalink]

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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 $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Answer: C

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 V
Joined: 02 Sep 2009
Posts: 55228
Re: M11-24  [#permalink]

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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 $$|F(x + 1)| = |(x + 1) - 1| = |x|$$

Answer: C

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 $$|F(x + 1)| = |(x + 1) - 1| = |x|$$.
_________________
Intern  B
Joined: 11 Feb 2017
Posts: 3
Re: M11-24  [#permalink]

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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  B
Joined: 07 Apr 2018
Posts: 1
Re: M11-24  [#permalink]

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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   [#permalink] 15 Oct 2018, 11:10
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