Domain Restrictions

Algebra 1 · Axiom Academy

Why a denominator can never be zero — and how to find the exact x -values an expression is forbidden to take. 1. Why a Zero Denominator Breaks Everything Watch what happens to as the denominator n shrinks toward 0 . The bar on the right is the size of the quotient — and it does not settle on an answer. It runs away to infinity. There is no number could possibly equal, so we call it undefined . There is also an algebraic reason. If equalled some number q , then multiplying both sides by 0 would force . But , a contradiction. So no such q exists. 2. Set the Denominator Equal to Zero The domain is every value x is allowed to be. A domain restriction is a value it can't be, because it would make the expression undefined. To find one, ignore the top entirely and ask: what makes the bottom zero? The animation lifts the denominator out, sets it equal to 0 , and solves. 3. Punching the Value Out of the Number Line A domain restriction literally removes a point from the number line of allowed inputs. For , every real number works except x = 3 . Watch a marker glide along the line and fall through the open hole exactly at 3 — that single point is missing from the domain. read aloud: "all real x except 3 " 4. At the Restriction, the Graph Blows Up Graph and something dramatic happens at x = 3 : the curve shoots off toward on one side and on the other. It approaches the dashed vertical line x = 3 but never touches it. That line is a vertical asymptote .

This is the written version of the interactive lesson above. See the full Algebra 1 course.