Precalculus Connections Web

Pre-Calculus · Axiom Academy

SUMMARY Pre-Calculus Connections Web How every major Pre-Calculus topic links to the others — and where each one leads in calculus. A function f assigns one output to each input — the single object that ties together every family below: polynomials, rationals, exponentials, logs, and trig functions are all just functions of different shapes. Inverses come in matched pairs: exponential logarithm, and trig inverse trig. An inverse undoes the original, so f^ -1 (f(x)) = x . Sequences and series let us add up infinitely many terms: a geometric series with converges to a single finite number. Vectors carry magnitude and direction; polar coordinates and complex numbers give the same point a new address. Every topic is a runway: the limit is where Pre-Calculus ends and calculus begins. The language of the whole course. A function pairs each input with exactly one output, and a one-to-one function has an inverse that reverses the pairing. Watch for: domain restrictions — division by zero and even roots of negatives carve gaps out of the domain. Leads to: limits, derivatives, and integrals all act on functions. Building Block Polynomials & Rationals Polynomials are sums of power terms; rationals are ratios of two of them. Rationals introduce vertical asymptotes wherever the denominator hits zero. When to use: modeling smooth curves (polynomials) or behavior that blows up or levels off (rationals). Leads to: asymptotes preview limits at infinity and indeterminate forms.

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