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M11-24

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Bunuel
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M11-24 [#permalink] New post   16 Sep 2014, 00:45
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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 statements must be true?

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. None of the above.
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C
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Bunuel
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Re M11-24 [#permalink] New post   16 Sep 2014, 00:45
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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 statements must be true?

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. None of the above.


I. \(f(|x|) = |f(x)|\).

This option is not necessarily true. Consider \(x = -1\). \(f(|x|) = f(1) =1 - 1 =0\) does not equal \(|f(x)| =|f(-1)| = |-1-1|= |-2|= 2\).

II. \(f(x) = f(-x)\).

This option is not necessarily true. Consider \(x = 1\). \(f(x) =f(1) = 1 - 1 = 0\) does not equal \(f(-x) = f(-1) = -1 - 1 = -1 - 1 = -2\).

III. \(|f(x + 1)| = |x|\).

This option 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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Bunuel
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Re: M11-24 [#permalink] New post   07 May 2023, 05:32
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
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Re M11-24 [#permalink] New post   09 Aug 2023, 11:04
I think this is a high-quality question and I agree with explanation.
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M11-24 [#permalink] New post   27 Dec 2023, 07:25
KarishmaB when we open functions of each statement, do we open them for the range of x or for the range of given function?

Ex.Which is the right wa to define these functions and hw do we know
for f(x+1) -{1-x-1 =-x ---- for x>=0
----------{1+x-1=x -----for x <0


or

for f(x+1) -{1-x-1 =-x ---- for x+1>=0
----------{1+x-1=x -----for x +1<0

Which of the domains do we define these functions for and why?
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Re: M11-24 [#permalink] New post   27 Dec 2023, 22:58
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Elite097 wrote:
KarishmaB when we open functions of each statement, do we open them for the range of x or for the range of given function?

Ex.Which is the right wa to define these functions and hw do we know
for f(x+1) -{1-x-1 =-x ---- for x>=0
----------{1+x-1=x -----for x <0


or

for f(x+1) -{1-x-1 =-x ---- for x+1>=0
----------{1+x-1=x -----for x +1<0

Which of the domains do we define these functions for and why?


Fair enough question. Take your cue from something you already know - absolute value functions

|x| = x when x >= 0
|x| = -x when x < 0

What is |x - 4|? It is (x - 4) when (x - 4) >= 0 i.e. when x >= 4
and -(x - 4) when (x - 4) < 0 i.e. when x < 4

Then, you must use "for x+1>=0 and for x +1<0" though it has no impact on this question's solution.
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Re M11-24 [#permalink] New post   03 Apr 2024, 05:41
I think this is a high-quality question and I agree with explanation.
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Re M11-24 [#permalink]
03 Apr 2024, 05:41
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