Question:
The graph of y = f(x) is given. The number of distinct zeroes of y = f(x) is :
The graph of y = f(x) is given. The number of distinct zeroes of y = f(x) is :
Show Hint
When counting the zeroes of a polynomial from its graph, only count the points where the graph meets the \(x\)-axis (horizontal axis).
Do not count the intersection point with the \(y\)-axis (vertical axis), as that represents the value \(f(0)\), not a zero of the function!
Do not count the intersection point with the \(y\)-axis (vertical axis), as that represents the value \(f(0)\), not a zero of the function!
Updated On: Jul 7, 2026
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Question:
We are given a graphical curve of a function \(y = f(x)\) plotted in a Cartesian coordinate system. We need to find the total number of distinct zeroes of the function \(f(x)\).
Step 2: Key Formula or Approach:
1. A "zero" of a function \(y = f(x)\) is any value of \(x\) for which the function's output is zero (\(f(x) = 0\)).
2. On a Cartesian graph, this is represented by the points where the curve meets (either intersects or is tangent to) the horizontal \(x\)-axis.
3. Therefore, the number of distinct zeroes is equal to the number of distinct locations on the \(x\)-axis where the curve intersects or touches it.
Step 3: Detailed Explanation:
1. Let us inspect the given graphical curve of \(y = f(x)\):
- The curve crosses from the third quadrant to the second quadrant, intersecting the \(x\)-axis at a negative value marked as point \(A\) (to the left of the origin \(O\)). This point of intersection is the first distinct zero of the function.
- The curve then rises to a peak in the upper-half plane, turns downwards, crosses the vertical \(y\)-axis, and goes down towards the positive \(x\)-axis.
- On the positive side of the \(x\)-axis (to the right of the origin \(O\)), the curve comes down, touches (is tangent to) the \(x\)-axis, and then turns back upwards into the first quadrant. This point of tangency represents a zero of even multiplicity. This is the second distinct zero.
2. Count the total number of distinct points on the \(x\)-axis:
- Point 1: At point \(A\) (to the left of the y-axis).
- Point 2: On the positive \(x\)-axis (point of tangency).
3. There are exactly 2 distinct locations where the curve meets the \(x\)-axis.
Thus, the number of distinct zeroes of \(y = f(x)\) is 2.
Step 4: Final Answer:
The number of distinct zeroes of the function \(f(x)\) is 2, which corresponds to option (C).
We are given a graphical curve of a function \(y = f(x)\) plotted in a Cartesian coordinate system. We need to find the total number of distinct zeroes of the function \(f(x)\).
Step 2: Key Formula or Approach:
1. A "zero" of a function \(y = f(x)\) is any value of \(x\) for which the function's output is zero (\(f(x) = 0\)).
2. On a Cartesian graph, this is represented by the points where the curve meets (either intersects or is tangent to) the horizontal \(x\)-axis.
3. Therefore, the number of distinct zeroes is equal to the number of distinct locations on the \(x\)-axis where the curve intersects or touches it.
Step 3: Detailed Explanation:
1. Let us inspect the given graphical curve of \(y = f(x)\):
- The curve crosses from the third quadrant to the second quadrant, intersecting the \(x\)-axis at a negative value marked as point \(A\) (to the left of the origin \(O\)). This point of intersection is the first distinct zero of the function.
- The curve then rises to a peak in the upper-half plane, turns downwards, crosses the vertical \(y\)-axis, and goes down towards the positive \(x\)-axis.
- On the positive side of the \(x\)-axis (to the right of the origin \(O\)), the curve comes down, touches (is tangent to) the \(x\)-axis, and then turns back upwards into the first quadrant. This point of tangency represents a zero of even multiplicity. This is the second distinct zero.
2. Count the total number of distinct points on the \(x\)-axis:
- Point 1: At point \(A\) (to the left of the y-axis).
- Point 2: On the positive \(x\)-axis (point of tangency).
3. There are exactly 2 distinct locations where the curve meets the \(x\)-axis.
Thus, the number of distinct zeroes of \(y = f(x)\) is 2.
Step 4: Final Answer:
The number of distinct zeroes of the function \(f(x)\) is 2, which corresponds to option (C).
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