Question:
Which of the following is not true about the graph of \(f(x)\)?
Which of the following is not true about the graph of \(f(x)\)?
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When analyzing graph behavior, split the function into piecewise components, analyze each segment separately, and determine quadrant presence using sign combinations.
Updated On: Jul 29, 2025
- A portion of the graph is parallel to the line \(y = 25\)
- A portion of graph is in 2nd quadrant.
- Some portion of graph lies in 1st quadrant.
- Some portion of graph lies in 3rd quadrant.
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The Correct Option is D
Solution and Explanation
Step 1: Understand the nature of the function
We are referring to the same function from Question 20: \[ f(x) = |x - 1| - x \] Break it into cases: For \(x \leq 1\), \[ f(x) = (1 - x) - x = 1 - 2x \Rightarrow \text{Linear decreasing line} \] For \(x>1\), \[ f(x) = (x - 1) - x = -1 \Rightarrow \text{A horizontal line at } y = -1 \] Step 2: Analyze quadrant locations
- For \(x <0\), \(f(x) = 1 - 2x>1\): so graph lies in 2nd quadrant. - For \(0 <x <0.5\), \(f(x)>0\): lies in 1st quadrant. - For \(x>1\), \(f(x) = -1\): line lies below x-axis, and for \(x>1\), both x and y are positive and negative → graph lies in 4th quadrant, not in 3rd. - 3rd quadrant requires \(x <0\) and \(y <0\). For \(x <0\), \(f(x) = 1 - 2x>1\), so never negative ⇒ does not enter 3rd quadrant. \[ \text{Hence, the graph does not lie in the 3rd quadrant.} \]
We are referring to the same function from Question 20: \[ f(x) = |x - 1| - x \] Break it into cases: For \(x \leq 1\), \[ f(x) = (1 - x) - x = 1 - 2x \Rightarrow \text{Linear decreasing line} \] For \(x>1\), \[ f(x) = (x - 1) - x = -1 \Rightarrow \text{A horizontal line at } y = -1 \] Step 2: Analyze quadrant locations
- For \(x <0\), \(f(x) = 1 - 2x>1\): so graph lies in 2nd quadrant. - For \(0 <x <0.5\), \(f(x)>0\): lies in 1st quadrant. - For \(x>1\), \(f(x) = -1\): line lies below x-axis, and for \(x>1\), both x and y are positive and negative → graph lies in 4th quadrant, not in 3rd. - 3rd quadrant requires \(x <0\) and \(y <0\). For \(x <0\), \(f(x) = 1 - 2x>1\), so never negative ⇒ does not enter 3rd quadrant. \[ \text{Hence, the graph does not lie in the 3rd quadrant.} \]
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