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 means on a graph.
A zero of $y = f(x)$ is a value of $x$ where the graph meets the x axis, since $f(x) = 0$ there.
So instead of reading the curve region by region, we scan the graph once from left to right and mark every point where the curve touches the x axis.
Step 2: Use the crossing vs touching rule.
A curve can meet the x axis in two ways. It can cross straight through, going from below the axis to above it or the other way round. It can also just touch the axis and turn back without crossing, which happens at a zero of even multiplicity.
Both a crossing point and a touching point count as one zero each, since $f(x) = 0$ at both.
Step 3: Sweep the graph left to right.
Starting from the far left of the curve, the first place it meets the x axis is a single crossing point to the left of the origin. This is zero number 1.
The curve then moves up into the region above the axis, comes back down on the right side of the origin, and this time it just touches the axis and turns back upward without crossing. This touching point is zero number 2.
No other point on the curve meets the x axis.
Step 4: Count the marks.
The sweep gave exactly two marks on the x axis, so the function has exactly 2 distinct zeroes.
Final Answer:
The number of distinct zeroes of $f(x)$ is 2, which corresponds to option (C).
\[ \boxed{2} \]
