Domains of composite functions

Domains of composite functions

Find the real domain of a composite function by checking both stages of the input-output chain. An input must first be allowed by the inner function, and the resulting output must then be allowed by the outer function. This practice set turns that principle into a dependable method for algebraic compositions.

Problems include square roots of linear, quadratic and rational expressions, a reciprocal applied to a square root, nested roots and functions with explicitly restricted domains. Several items simplify to a familiar expression while retaining exclusions inherited from the original composition. Another compares the domains obtained when the order of two functions is reversed.

Before beginning, review interval notation, quadratic inequalities, denominator restrictions and the meaning of composition. List the inner restriction before solving the outer condition. Check boundary values explicitly to decide whether an endpoint is included; do not decide from the appearance of a simplified formula alone.

Single-select and multiselect questions progress from direct conditions to intersections, sign reasoning and explanation. Detailed solutions show why the two checks are separate and why the intersection of the two original domains is generally not the correct composite domain. All variables and functions are real-valued unless the question states an explicit smaller domain.