Rational Function Definition

Pre-Calculus · Axiom Academy

LESSON Rational Function Definition A rational function is just one polynomial divided by another — and the denominator's zeros are the forbidden inputs. A rational function is anything you can write as one polynomial over another. Watch the numerator P(x) and the denominator Q(x) click into place above and below the fraction bar. where P(x) and Q(x) are polynomials and 2. What Counts — and What Doesn't To qualify, both the top and the bottom must be polynomials — sums of whole-number powers of x with coefficients. A square root, an absolute value, or an exponential anywhere disqualifies it. The animation sends each candidate through a gate that passes the polynomials and rejects the rest. Every polynomial is secretly rational A plain polynomial like x^2+3x+2 counts too: write it as . The constant 1 is a perfectly good (degree- 0 ) denominator, so every polynomial is a rational function with denominator 1 . 3. Where the Denominator Hits Zero Set the denominator equal to zero and solve — those x -values are forbidden , because dividing by zero is undefined. For the simplest rational function the denominator is x , which is zero only at x=0 . Watch the real graph: as x slides toward 0 , the curve rockets away from the forbidden line. 4. A Bigger Denominator, More Forbidden Inputs

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