One-to-one functions and domain restriction

One-to-one functions and domain restriction

Determine when an output uniquely identifies its input. This precalculus practice set develops one-to-one reasoning through a finite table, a coordinate graph, algebraic proofs and counterexamples. It distinguishes being a function from being a function whose outputs never repeat at different inputs.

Begin with the definition and horizontal-line test, then study how a carefully chosen domain changes the conclusion. Questions include a maximal interval for a shifted parabola, a finite domain containing positive and negative values, a constant function on a singleton, and a time-based measurement that becomes unambiguous on one part of the day.

Review graph behavior, symmetry and function domains before starting. To disprove a one-to-one claim, exhibit two allowed inputs with the same output. To prove it, use the definition or a justified strict-monotonicity argument. A few sample values alone cannot establish a global property.

Single-select and multiselect items progress toward distinctions among strict increase, weak monotonicity, even or odd symmetry and injectivity. The questions also distinguish the range actually reached from a larger proposed output set. Detailed explanations prepare you for finding inverse functions while keeping the focus here on whether a reversal can assign a unique input.