Question:
Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as:
a) if \(f(x)=3f(-x)\)
b) if \(f(x)=-f(-x)\)
c) if \(f(x)=f(-x)\)
d) if \(3f(x)=6f(-x)\), for \(x\ge0\)
Mark the option that applies to the graph of this question (shown in the figure).
Given below are three graphs made up of straight-line segments shown as thick lines. In each case choose the answer as:
a) if \(f(x)=3f(-x)\)
b) if \(f(x)=-f(-x)\)
c) if \(f(x)=f(-x)\)
d) if \(3f(x)=6f(-x)\), for \(x\ge0\)
Mark the option that applies to the graph of this question (shown in the figure).
a) if \(f(x)=3f(-x)\)
b) if \(f(x)=-f(-x)\)
c) if \(f(x)=f(-x)\)
d) if \(3f(x)=6f(-x)\), for \(x\ge0\)
Mark the option that applies to the graph of this question (shown in the figure).
Show Hint
Look at the symmetry of the graph about the origin: compare the left piece with the right piece.
Updated On: Nov 4, 2026
- if \(f(x)=3f(-x)\)
- if \(f(x)=f(-x)\)
- if \(f(x)=-f(-x)\)
- if \(3f(x)=6f(-x)\), for \(x\ge0\)
Show Solution
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The Correct Option is C
Solution and Explanation
Step 1: Read the graph.
The graph is flat (\(f=0\)) from \(x=-4\) to \(x=4\). For \(x>4\) it falls in a straight line (slope about \(-0.75\)) to about \(-4.5\) at \(x=10\). For \(x<-4\) it rises as one goes left, reaching about \(+4.5\) at \(x=-10\), a mirror image of the right part through the origin.
Step 2: Test the symmetry.
A point \((x,\,y)\) on the right part has a matching point \((-x,\,-y)\) on the left part (for example \((10,-4.5)\) and \((-10,4.5)\)); on the flat middle part \(f=0\) on both sides. So \(f(-x)=-f(x)\) for all \(x\), i.e. \(f(x)=-f(-x)\): the graph is symmetric about the origin (an odd function).
Step 3: Rule out the others.
At \(x=10\): \(f(10)\approx-4.5\) and \(f(-10)\approx4.5\). Then (a) would need \(-4.5=13.5\), (c) would need \(-4.5=4.5\), and (d) would need \(3f(10)=-13.5\) to equal \(6f(-10)=27\): all false.
Final Answer:
The correct option is (3): if \(f(x)=-f(-x)\)
The graph is flat (\(f=0\)) from \(x=-4\) to \(x=4\). For \(x>4\) it falls in a straight line (slope about \(-0.75\)) to about \(-4.5\) at \(x=10\). For \(x<-4\) it rises as one goes left, reaching about \(+4.5\) at \(x=-10\), a mirror image of the right part through the origin.
Step 2: Test the symmetry.
A point \((x,\,y)\) on the right part has a matching point \((-x,\,-y)\) on the left part (for example \((10,-4.5)\) and \((-10,4.5)\)); on the flat middle part \(f=0\) on both sides. So \(f(-x)=-f(x)\) for all \(x\), i.e. \(f(x)=-f(-x)\): the graph is symmetric about the origin (an odd function).
Step 3: Rule out the others.
At \(x=10\): \(f(10)\approx-4.5\) and \(f(-10)\approx4.5\). Then (a) would need \(-4.5=13.5\), (c) would need \(-4.5=4.5\), and (d) would need \(3f(10)=-13.5\) to equal \(6f(-10)=27\): all false.
Final Answer:
The correct option is (3): if \(f(x)=-f(-x)\)
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