Combined rational and radical restrictions

Combined rational and radical restrictions

Determine domains and ranges when rational expressions and radicals appear together. This precalculus practice develops a systematic way to combine nonnegative radicands, nonzero denominators and restrictions inherited from an original expression. It builds directly on rational inequalities and radical functions.

Questions include a rational expression under a square root, a root in a denominator, an external denominator zero, cancellation inside a radical and nested roots. Range tasks ask you to distinguish outputs that are approached from outputs that are attained, and to use a restricted input interval correctly.

Prerequisites are interval notation, sign charts, real principal roots and function composition. Single-select and multiselect tasks progress from direct restrictions to combining several conditions and proving that an output is missing. No calculator or calculus is required.

Do not apply one blanket rule to every denominator: a square root in a denominator requires a strictly positive radicand, while a separate denominator may remove a point from an otherwise valid root domain. Work from the inside outward for nested functions, and preserve exclusions through simplification. Detailed explanations show how each boundary enters the answer and why a domain conclusion cannot simply be copied as a range. These habits also prepare you to track the two-input restrictions in difference quotients.