Domain of Rational Functions

Pre-Calculus · Axiom Academy

LESSON Domain of Rational Functions A rational function lives everywhere on the real line except where its denominator hits zero — so finding the domain means hunting down those forbidden points. 1. The One Rule: Never Divide by Zero Take any rational function . As long as the denominator Q(x) is nonzero, the fraction is a perfectly ordinary number. But at any x where Q(x)=0 , the output explodes — the function is undefined there, and that single point is kicked out of the domain. Domain = every real number that survives Remove exactly the zeros of the denominator To find the domain, ask which x make the denominator zero, then exclude them. Here only the denominator matters — the numerator can be anything. Example 1 — find the domain of 3. When Two Points Are Forbidden If the denominator is a higher-degree polynomial, it can vanish at several places. Factor it , set each factor to zero, and you get every forbidden x at once. Example 2 — find the domain of 4. Three Ways to Write the Domain Once you know which points to remove, you can record the surviving set in three equivalent ways. Using Example 1 ( ): Describe the set by a condition. Glue the surviving pieces with a union. Sometimes the denominator has no real zeros at all. Then there is nothing to punch out, and the domain is the entire real line. Example 3 — find the domain of

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