Question

The graph of $y = f(x)$ is given. The number of zeroes of $f(x)$ is :

• A. 0
• B. 1
• C. 2
• D. 4

Hint:

Always look carefully at which axis is being intersected.
The zeroes of $y = f(x)$ are the intersections with the x-axis.
If the equation were $x = f(y)$, the zeroes would be the intersections with the y-axis.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks us to find the number of zeroes of the polynomial function $f(x)$ using its graphical representation on the coordinate plane.
A zero of a polynomial $f(x)$ is a value of $x$ for which $f(x) = 0$.

Step 2: Key Formula or Approach:
Geometrically, the zeroes of a polynomial $y = f(x)$ are the x-coordinates of the points where the graph intersects or touches the x-axis.
Thus, we simply need to count the total number of intersection points of the curve with the horizontal x-axis ($X'OX$).

Step 3: Detailed Explanation:

• Let us examine the horizontal line representing the x-axis in the given graph.

• Now, trace the continuous curve $y = f(x)$ from the far left to the far right.

• As we move along the curve, we observe that:
- The curve crosses the x-axis once on the negative side (left of the origin O).
- It then goes down, turns, and crosses the x-axis again near the origin.
- It reaches a peak, turns downwards, and crosses the x-axis a third time on the positive side.
- It goes down, turns upwards, and crosses the x-axis a fourth time further to the right.

• Counting these distinct intersection points, we find that there are exactly 4 points where the curve crosses the horizontal line $y = 0$.

• Therefore, the polynomial equation $f(x) = 0$ has 4 real roots, which correspond to the 4 zeroes of $f(x)$.


Step 4: Final Answer:
The number of zeroes of the polynomial $f(x)$ is 4.
Hence, option (D) is correct.