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!