Maths: From Cluttered Fraction to Clear Graph

Rational functions can hide surprising structure beneath their fractional form. A function like f(x) = (x² − 4)/(x² − x − 6) may look complicated, yet factorising numerator and denominator often reveals a common factor that cancels, simplifying the expression dramatically. This process works because polynomials share roots in predictable ways: x² − 4 becomes (x − 2)(x + 2), while x² − x − 6 becomes (x − 3)(x + 2). Cancelling (x + 2) is valid only where it is nonzero, so the domain excludes that value. Understanding this matters because discontinuities and asymptotes define a function's true behaviour. After cancellation, the remaining denominator reveals where vertical asymptotes occur, while comparing leading terms shows the horizontal limiting value as x grows large. Each step — factorising, cancelling, then analysing the simplified form — connects algebra to the graph's shape, turning a messy expression into a clear picture of how the function behaves across its domain.