Maths: How Factoring Exposes Rational Discontinuities

Discontinuities in rational functions reveal where an expression breaks down and why. A rational function is undefined wherever its denominator equals zero, so factoring expressions like x² − 1 = (x − 1)(x + 1) exposes the excluded values that form the domain. These values matter because they mark the boundary between valid inputs and points where the function's behaviour changes fundamentally. Not every excluded value behaves the same way. Cancelling common factors after factorisation can simplify the expression, and whether the simplified form remains defined at that point determines the nature of the discontinuity. If it does, the graph has a hole; if the denominator still vanishes, a vertical asymptote appears. Combining fractions over a common denominator often reveals this simplified structure, showing how domain restrictions, algebraic simplification, and graphical behaviour are all connected.