Rational Functions

Rational functions divide polynomials and may have holes, asymptotes, intercepts, and separate branches.

Excluded inputs appear as breaks or vertical asymptotes. Two branches approach dashed horizontal and vertical asymptotes without touching them. The separation emphasizes that denominator restrictions belong to both the algebra and the graph.

Factoring a rational function reveals where its graph is missing, where it grows without bound, and where it crosses the axes.

Definition and domain

A rational function has form

f(x)=\frac{P(x)}{Q(x)},qquad Q(x)\ne0,

where $P$ and $Q$ are polynomials. The domain excludes every real zero of the original denominator.

These exclusions can create vertical asymptotes or removable discontinuities called holes.

Worked example: full feature analysis

Sign and interval behaviour

Zeros and vertical asymptotes divide the domain into intervals. A sign chart determines where the function is positive or negative. Near a vertical asymptote, checking each side separately reveals whether the graph approaches $+\infty$ or $-\infty$.

One-sided behaviour matters because the two sides may differ.

Solving intersections

To find where a rational function meets another function, set the formulas equal, note restrictions, clear denominators, and solve. Any candidate excluded from the original domain must be rejected.

A graph can verify the number and approximate location of intersections.

Common mistakes

Calling every excluded value a vertical asymptote. Cancelled factors create holes.

Using the original numerator for intercepts after cancellation. A cancelled zero is missing, not an intercept.

Treating a horizontal asymptote as uncrossable. It describes end behaviour.

Forgetting the hole's $y$-coordinate. Substitute into the simplified formula.

Graphing across a vertical asymptote as one connected curve. The domain is split into branches.

Quick self-check

What are the original domain restrictions?
Which factors cancel and which remain below?
Where are holes and vertical asymptotes?
What end behaviour follows from degree or division?
Which intercepts survive the restrictions?
Does the graph's sign and branch behaviour match test values?