Functions and Their Graphs

GRE Quantitative · Axiom Academy

LESSON Functions and Their Graphs A function is a machine: feed in a number, get one number back — and its graph is just a picture of every input/output pair it produces. 1. Function Notation — the Machine at Work f(x) means "the output of the machine f when you feed it x ." Watch the point slide along the input axis — for every x it drops straight up or down onto the curve, and that landing height is f(x) . The curve isn't a separate object to memorize; it's just every output, plotted . 2. Domain and Range — What the Machine Accepts and Returns Not every machine accepts every input. Domain = the full set of x -values you're allowed to feed in. Range = the full set of outputs that come back out. Watch the sweep below: it only travels across the domain, and the shaded band on the output axis only ever covers the range — nothing outside it is ever produced. Negative numbers have no real square root, so this machine simply refuses them. — the sweep never crosses left of zero. — a principal square root is never negative, so the shaded output band never dips below zero either. f(x) = x^2-2x-3 accepts every real number (domain = all reals) but its lowest output is -4 at the vertex, so its range is . On the GRE, domain is read left-to-right along the x-axis (where does the curve exist?) and range is read bottom-to-top along the y-axis (how low/high does it reach?) — you rarely need algebra at all once you can see the picture. 3. Composition — Wiring Two Machines Together

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