Graphs Tell the Story

Every function can be graphed by plotting input-output pairs. The function f(x) = x squared produces a parabola — a U-shaped curve. f(x) = 1/x produces a hyperbola. The shape of the graph reveals the function's behavior: growth, decay, symmetry, and limits all become visible at a glance.

The vertical line test determines whether a graph represents a function: if any vertical line crosses the graph more than once, it's not a function. A circle fails this test — for some x values, there are two y values. Functions demand one output per input, no exceptions.