The Horizontal Line Test

Pre-Calculus · Axiom Academy

LESSON The Horizontal Line Test A quick visual test that tells you, at a glance, whether a function is one-to-one — and therefore whether it has an inverse. 1. One Output, Two Inputs — the Problem A function is one-to-one when each output comes from exactly one input: no two different x -values are allowed to land on the same y -value. That is the precise condition an inverse needs. One-to-one: equal outputs force equal inputs A horizontal line is the set of all points at one height — one fixed y . So if such a line crosses the graph twice , two different x 's produce that same y , and the function is not one-to-one. Watch the line sweep down f(x)=x^2 : at every positive height it solves and lands on both arms at once. Here is the rule in full. It is an "if and only if" — it detects one-to-one functions exactly, with no exceptions. Contrast the parabola with f(x)=x^3 . Solving x^3=y gives the single real root , so a horizontal line at any height meets the cubic exactly once. Watch the same sweep — the crossing count never leaves 1 . Every horizontal line touches the graph at most once, so each output owns a single input. The cubic only ever climbs, so it can never revisit a y -value — a clean way to be one-to-one. Pass the test and the curve can be read backwards: here . A single line with two crossings — like the parabola's — is enough to disqualify the whole function. Why "strictly monotonic" is a shortcut

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