Functions Summary

Pre-Calculus · Axiom Academy

SUMMARY Functions and Their Graphs Everything this unit taught about reading a function — its inputs, its shape, its symmetry, and how fast it changes — in one place. A function pairs each input with exactly one output; its domain is every legal input and its range is every output it actually produces. A graph's shape tells the story: where it rises or falls (increasing / decreasing) and where it turns (relative maxima and minima). Symmetry is an algebra test, not a guess: f(-x)=f(x) means even (mirror over the y -axis), f(-x)=-f(x) means odd (rotate about the origin). Piecewise functions switch rules by interval — always check which piece an input lands in before you evaluate. The average rate of change is the slope of a secant line; pushing the two points together via the difference quotient is the doorway to calculus. The domain is every input the rule accepts; the range is every output it can return. Two red flags shrink a domain: a denominator that hits zero, or a negative quantity under an even root. Watch out for: excludes x=2 ; needs . From a graph: read the domain off the x -axis shadow, the range off the y -axis shadow. Core Concept Increasing & Decreasing Read a graph left to right. Where it climbs, the function is increasing ; where it falls, it is decreasing . These are always stated as intervals of x , never single points. Watch out for: describe behavior over open intervals of inputs, not outputs.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.