Domain of Polynomial and Rational Functions (F 508)

A polynomial is just adding, subtracting, and multiplying x by itself and by numbers. No step in that process ever breaks, no matter what real number you plug in, so a polynomial's domain is all real numbers. Division is different: dividing by zero is undefined. So when a polynomial sits in a denominator, you have to find every x that would zero it out and remove those from the domain. Square roots bring their own rule: you can't take the square root of a negative number and get a real output, so whatever is under the root sign has to stay zero or positive.

The mechanism for a rational function is mechanical on purpose: take the denominator, set it equal to zero, and solve. Every solution gets excluded; everything else is fair game. For f(x) = (x+3)/(x²-5x+6), factor the bottom into (x-2)(x-3), so x = 2 and x = 3 are excluded and the domain is all reals except those two. For g(x) = √(2x-7), solve 2x-7 ≥ 0 to get x ≥ 3.5, so the domain starts at 3.5 and the graph begins there, moving right.

This connects directly to vertical asymptotes and holes, which sit right next to domain in this unit. If a factor cancels between numerator and denominator, that input still gets excluded from the domain, but it shows up as a hole in the graph instead of a vertical asymptote. Domain is the gatekeeper: once you know which inputs are allowed, you can start asking what the graph actually does at the edges.