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:

Do not confuse intersections with the $y$-axis as zeroes.
Only count the intersection points on the horizontal $x$-axis.
An intersection with the $y$-axis represents the value of $f(0)$, not the zeroes of the function.

Answer 1

The Correct Option is C

The number of zeroes of a function shown as a graph is exactly the number of times the curve cuts or touches the x-axis, since a zero is a value of $x$ where $y = f(x) = 0$. Instead of walking through the graph feature by feature, let's rule out each wrong option first.

1. 0: This would mean the curve never touches the x-axis anywhere on the drawn graph. That is not the case here, the curve is drawn crossing the horizontal axis, so a count of zero zeroes is too low.
2. 1: This would require the curve to touch the x-axis at exactly one point, either just grazing it or crossing it once and then moving away without returning. The given graph crosses the axis more than once, so this option undercounts the crossings.
3. 2: The curve dips down, meets the x-axis, comes back up, and meets the x-axis again, giving exactly two distinct meeting points with the horizontal axis. This matches the picture given in the question.
4. 4: This would need four separate crossings, which means the curve would have to change direction across the axis twice as often as it actually does in the given figure. The graph shown does not wiggle that many times, so 4 is too high.

Since the curve meets the x-axis at exactly two points and nowhere else, the number of zeroes of $f(x)$ is 2, which is option (C).

Let's summarize:

• A zero of $f(x)$ is a point where the graph meets the x-axis.
• Counting the meeting points directly on the given curve gives exactly two zeroes.

So the number of zeroes of $f(x)$ is 2.