Analyzing a Rational Function

Analyzing a rational function means factoring both floors, then reading the domain, the holes, the vertical asymptotes, the horizontal asymptote, and the intercepts — in that order, off the factors.

Analyze r(x) = (x² − 9)/(x² + 5x + 6) = ((x − 3)(x + 3))/((x + 2)(x + 3)). Domain: all reals except −2 and −3. The canceled factor x + 3 leaves a hole at (−3, 6); the surviving factor x + 2 raises a vertical asymptote at x = −2. Equal degrees put the horizontal asymptote at y = 1. Intercepts from the simplified form: x-intercept where the numerator zeroes, (3, 0); y-intercept r(0) = −9/6 = −3/2. Six features, all read from one factorization — and the hole-versus-asymptote call is the whole game: canceled factors leave holes, surviving factors leave walls.