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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.
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.
16 Sep 2014, 00:45
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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 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
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
General Discussion
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.
