Special Functions: Piecewise, Floor and Absolute Value

Most functions follow a single rule everywhere. These three do not. Each one either changes its rule partway along, chops a number downward, or strips away a minus sign — and each leaves a distinctive shape on the graph.

Concept The piecewise function

A piecewise function uses a different rule depending on which interval x falls into.

Read the conditions carefully. Here belongs to the second rule, because that condition uses "greater than or equal to". Every input must match exactly one case.

Example Evaluating a piecewise function

Using the function above, find and .

−3 is less than 0, so use

0 satisfies

, so use

⟹ f(−3) = −1 and f(0) = 0

Concept The floor function

The floor function

gives the largest integer that is less than or equal to x. It always moves down — it never rounds to the nearest whole number.

— note that rounding would give 2.

Think of a teacher who simply discards the fractional part of a mark. Each step begins with a filled circle and ends with a hollow one, because the next whole number belongs to the following step.

Concept The absolute value function

The absolute value

turns any negative input positive and leaves positive inputs untouched. The two halves meet at the origin, producing a sharp V.

and

Notice that absolute value is itself a piecewise function — the first idea in this lesson, applied to a particular pair of rules.

Summary

A piecewise function changes its rule from one interval to the next.
Check which case a boundary value like x = 0 belongs to.
The floor function ⌊x⌋ takes the greatest integer not above x — always downward.
Absolute value |x| makes negatives positive and leaves positives alone.
Their graphs are separate pieces, a staircase, and a V.