Piecewise Functions • Topic 2 of 3

Graphs

On a graph, a piecewise function is drawn one piece at a time, each only over its own interval. A closed (filled) circle marks an endpoint that is included; an open circle marks one that is excluded. Where the two pieces do not meet, the graph has a jump, called a discontinuity; where they connect, the function is continuous. A step function, like a postage-rate chart, is a familiar piecewise graph made of flat segments. To read such a graph, find the input on the x-axis, see which piece covers it, and read the output. The SAT tests the meaning of open versus closed circles and recognising jumps.

A step function: four horizontal segments at increasing heights, each with a filled circle at its left end and an open circle at its right end.Step functionOOpen circle = not included; filled = included

✅ Solved examples

1. What does an open circle on a piecewise graph mean?
That endpoint is not included.
2. What does a filled (closed) circle mean?
That endpoint is included.
3. A break where the two pieces do not meet is called a:
Discontinuity (a jump).
4. A postage-rate chart drawn as flat steps is what kind of function?
A piecewise (step) function.

✏️ Practice — try these, take hints as needed

1. On a piecewise graph, a closed circle means the point is:
Filled = included?
—
—
Included.
2. An open circle on a piecewise graph means the point is:
Open = excluded?
—
—
Not included.
3. When the two pieces connect with no break, the function is:
No jump.
—
—
Continuous.
4. A graph made of flat horizontal segments at different levels is a:
Steps.
—
—
Step function.
5. A sudden vertical break between two pieces is a:
Gap in the graph.
—
—
Discontinuity (jump).

📝 Topic test — 8 questions

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