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!
Step 1: Recall what a zero of a graph means, using the factor idea.
A zero of $f(x)$ is a value of $x$ where the curve touches or crosses the horizontal axis. Every time the curve meets the $x$-axis, that $x$-value is a root of the equation $f(x) = 0$, and each root corresponds to a factor of $f(x)$ of the form $(x - k)$.
Step 2: Separate crossing points from touching points.
When a curve crosses straight through the $x$-axis, that root has odd multiplicity (like a simple factor $(x-k)$).
When a curve touches the $x$-axis and turns back without crossing, that root has even multiplicity (like a repeated factor $(x-k)^2$), but it is still counted as one distinct zero, since it is still one distinct $x$-value where $f(x) = 0$.
Step 3: Read the given graph with this rule.
The curve meets the $x$-axis at one point to the left of the origin where it crosses straight through, giving one distinct root.
It meets the $x$-axis again at one point to the right of the origin where it touches and turns back, giving a second distinct root (even multiplicity, but still one distinct location).
Step 4: Count the distinct locations, not the multiplicities.
The question asks for the number of distinct zeroes, not the total multiplicity, so we simply count the separate $x$-values where the graph meets the axis: that count is 2.
Step 5: Final Answer.
The number of distinct zeroes of $y = f(x)$ is 2, so option (C) is correct.
\[ \boxed{2} \]
