Ranges and attainable outputs

Find the outputs a function actually reaches, rather than merely listing possible bounds. This precalculus practice set develops range reasoning through finite data, restricted linear and quadratic rules, radicals, reciprocals and an original coordinate graph. Each question specifies the domain or uses the natural real domain of its formula.

Open and closed endpoints are central: a value can be approached without being attained. You will also see why removing one input need not remove its output, why cancellation can leave a constant output, and how a physical domain changes the possible values of an area model.

Prerequisites

Review domain restrictions, intervals, square roots and elementary quadratic equations. For a proposed output, try solving for an allowed input. Pair any lower or upper bound with a reason that all claimed outputs really occur.

How to practise

Single-select and select-all questions progress from direct interpretation to justification. Sketch a quick graph when useful, but verify endpoint inclusion in the original rule. Read the explanations after attempting each task, then write an input that attains an included endpoint or explain why an excluded one cannot occur. These steps distinguish a justified range from an estimate based only on appearance.

Ranges and attainable outputs