M11-24

I think the question should be change to "Which of the following MUST be True about f(x)?" to make it more clear.

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

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!

Please explain this in more detail

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?

I think this is a high-quality question and I don't agree with the explanation. If we consider x=0 for statement III, then it breaks.
Hence correct answer should be E not D I think.

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

Bunuel can you pls open st 1 and 2 the wy you have opened 3 in this thread? How do we find the intervals and accordingly expand it. The steps are not clear