Rational Functions Summary

Pre-Calculus · Axiom Academy

Everything you learned in this unit — structure, asymptotes, holes, graphing, and solving equations and inequalities. A rational function is a ratio of polynomials ; its domain excludes every value where q(x)=0 . Factor first. The factored form reveals every feature — vertical asymptotes, holes, and intercepts — at a glance. A factor that cancels gives a hole ; a leftover denominator zero gives a vertical asymptote . End behavior (the horizontal or oblique asymptote) is decided by comparing the degrees of p and q . When solving, clear the LCD and always check for extraneous solutions; for inequalities, use a sign chart with the denominator's zeros excluded. Core Concept Definition & Domain A rational function is a quotient of two polynomials. It is defined everywhere except where the denominator is zero, so the domain is all real numbers with those values removed. Domain: solve q(x)=0 and exclude those x . Watch out for: find the domain from the original denominator, before any cancelling. Core Concept Vertical Asymptotes After fully cancelling, a value that still makes the denominator zero (but not the numerator) gives a vertical asymptote — the graph shoots to there. When to use: on the simplified function, set the leftover denominator =0 . Watch out for: if the factor cancelled, it's a hole, not a V.A.

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