Rational domains, intercepts and holes

Rational domains, intercepts and holes

Analyze rational functions by preserving the original domain while simplifying their formulas. This precalculus practice focuses on removable holes, valid horizontal and vertical intercepts, and the distinction between a cancelled factor and a restored input.

Questions cover hole coordinates, missing intercepts, multiple excluded inputs, repeated factors and an original graph with an open point. Some cases have no holes; others have a hole on the horizontal axis or a zero numerator across the entire allowed domain. A parameter question checks what happens when a zero and an excluded input coincide.

Prerequisites are factoring, rational expressions, real domains and polynomial multiplicity. Single-select and multiselect tasks progress from reading features to reasoning about exceptional cases. All required information appears in each question, and the graph has readable axes and scales.

First identify excluded inputs from the original denominator. Then simplify, check whether any denominator factor remains at each excluded input, and evaluate the reduced expression only to locate a removable hole. Explanations show why a cancelled location is still absent from the graph. One-sided behavior and full asymptote analysis belong to later practice; this set builds the domain and intercept reasoning they require.