Piecewise functions and absolute-value graphs

Piecewise functions and absolute-value graphs

Learn how one function can use different formulas on different parts of its domain. This precalculus practice set connects branch conditions, open and closed graph endpoints, absolute value and threshold-based models. Original coordinate diagrams support questions about boundary values, recovered rules and the vertex of an absolute-value graph.

Applications include a fictional shipping tariff and a capped charge. Other tasks test whether overlapping branch conditions actually conflict, whether a formula leaves an input undefined, and whether a candidate solution belongs to the branch used to obtain it. The emphasis is careful reasoning about conditions; formal continuity and derivative tests are not required.

Prerequisites

Review function evaluation, real domains, inequalities and absolute value. First determine which input condition applies, then use that formula. When solving for an input, check the candidate against the branch condition as well as the equation.

Study approach

Work from the rule or diagram before reading the choices. Single-select and select-all questions progress toward parameter matching, model construction and a two-breakpoint absolute-value expression. Worked explanations identify the relevant interval, show the computation and explain why an attractive shortcut fails. Keep a sketch or number line beside your calculations to track endpoints.