Radical graphs, inverses and equations

Radical graphs, inverses and equations

Connect radical graphs with the restricted power functions they reverse. This precalculus practice emphasizes domain, range and branch choice rather than treating a square root as an unrestricted reversal of squaring. An original plotted radical curve supports graphical interpretation of an inverse.

Questions include decreasing square-root curves, restricted quadratics and fourth powers, odd-root inverses, folded graphs and intersections with a line. Parameter and geometric tasks extend the same reasoning to the existence of solutions and the physical domain of an inverse model.

Prerequisites are function transformations, inverse relationships, real square-root domains and checking solutions after squaring. Single-select and multiselect practice progresses from recognizing allowed outputs to coordinating algebra, graph information and restrictions. Exact answers are used, with no calculus required.

When reversing a function, interchange domain and range before choosing a sign. When solving an equation involving a principal square root, use its nonnegative output to screen candidates. Detailed explanations show why a negative branch can be essential for an inverse, why an odd root has different real-domain behavior, and why a squared equation can include an intersection that the original graphs do not share. These distinctions support later nested-domain and difference-quotient work.