One-one functions and the horizontal line test

5 minAS Level1.2.11.2.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can say what makes a function one-one, test a graph with the horizontal line test, and restrict a domain so that a function becomes one-one.

One locker, one owner

Every student gets exactly one locker: that's a function (student ↦ locker). Now ask the reverse question: does every locker belong to just one student?

If no two students share a locker, the function is one-one.
If two siblings share locker 42, it's many-one. Still a function, but you can't name "the" owner of locker 42.

Where the analogy breaks: a school has a finite number of lockers. A function's domain can be infinite.

Definition

One-one function

A function $f$ is one-one if each value in its range comes from exactly one value in its domain.

Another way to say it: if two inputs $a$ and $b$ give the same output, $f(a) = f(b)$ , then they must be the same input, $a = b$ .

A function that is not one-one is called many-one.

The horizontal line test

Sketch $y = f(x)$ over the given domain only, then imagine sliding a horizontal ruler up the page.

Some horizontal line meets the graph twice or more → two inputs share an output → many-one.
Every horizontal line meets it at most once → one-one.

(The vertical line test is a different question: whether the graph is a function at all.)

Quick check

Fill in the blanksQuestion 1 · Recall

Multiple choiceQuestion 2 · Recall

Answer the 2 questions above to continue.