Piecewise Functions

A piecewise function uses different formulas on different parts of the domain.

To evaluate, first find which interval your input falls in, then use that rule. For a function that is 2x + 1 when x < 3 and x² when x ≥ 3, evaluating at x = 5 uses x², giving 25.

To graph, draw each piece only over its own interval.

At the boundaries, use a closed dot where the endpoint is included (≤ or ≥) and an open dot where it is not (< or >). Each x-value gets exactly one closed dot, or it would not be a function.

Absolute value is secretly piecewise: |x| is x when x ≥ 0 and −x when x < 0.

A piecewise function is defined by cases whose domains partition the intended domain — they must not overlap, or the definition would assign two outputs to one input.

Evaluation requires checking the condition first; substituting into the wrong branch is the dominant error and is invisible in the arithmetic.

Continuity at a boundary c requires the one-sided values to agree: lim_{x→c⁻} f(x) = lim_{x→c⁺} f(x) = f(c). In Algebra 2 this is checked by substituting c into both surrounding rules and comparing. A mismatch produces a jump discontinuity, which is a legitimate function, not an error — step functions and tax brackets are modelled exactly this way.

Absolute value functions convert to piecewise form by splitting at the zero of the inner expression, which is what makes |2x − 6| graphable as a V with vertex at x = 3 and is the standard technique for solving equations that mix absolute values with other terms.