One-to-One Functions

Pre-Calculus · Axiom Academy

Functions where every output comes from exactly one input — and why only those can be reversed. Plot the real parabola y=x^2 . Slide a point out to x=-a and another to x=+a at the same time. Drop each one straight down to the y -axis — and they hit the same height a^2 . Two different inputs, one shared output: that is precisely what a one-to-one function forbids. -a and +a are different inputs, yet… …they share one output. Not one-to-one. Picture the inputs on the left and the outputs on the right, with an arrow for each input. A function is one-to-one when no two arrows ever land on the same target. On the right below, two inputs collide on one output (the rule breaks); on the left, every arrow finds its own private output. " " is just the careful way to say the only way to get the same output is to have started with the same input . The moment two distinct arrows merge, that implication fails. 3. A Function That Passes: y=x^3 Now plot the real curve y=x^3 . Send a horizontal level line up and down through it. At every height it crosses the curve exactly once — a single input for each output. Because x^3 only ever climbs (its slope 3x^2 is never negative), outputs can never repeat. slope is never negative → strictly increasing → one-to-one Why it works: if the curve only goes up, a taller output is always reached farther right, so two different inputs can never tie. Each of comes from one and only one of . Dips then rises — a level line can hit it twice. Not one-to-one.

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