Domains and restrictions

Domains and restrictions of real functions

Determine which inputs a real-valued function actually permits. This precalculus practice set covers rational expressions, square roots, cube roots, combined restrictions and domains read from a coordinate graph. You will preserve exclusions from an original expression even when algebraic simplification seems to remove them.

The questions distinguish a radical in a numerator from a radical in a denominator, examine sign conditions for a rational radicand, and combine several requirements by intersection. Contextual tasks also separate algebraically possible inputs from meaningful positive whole-number quantities. A graph with open endpoints and a gap requires reading the horizontal extent of the plotted function.

Prerequisites

Be comfortable with interval notation, linear and simple quadratic inequalities, factoring and real radicals. No formal composition of functions or calculus is required. Begin with the original rule and write each restriction before simplifying.

Practice strategy

Single-select and select-all items mix calculations with explanations of invalid shortcuts. Test proposed endpoints directly, identify forbidden denominator zeros, and verify a sample input in each claimed interval. The detailed feedback shows why individual conditions must hold simultaneously and why a domain answer includes every permitted input, rather than only a few working examples.