# Horizontal Line Test

How to use the horizontal line test to see if a function is one-to-one: how to, examples, and their solutions.

## One-to-One, How to Do

One-to-one means 'one *x*, one unique *y*'.

So if two *x* is mapped to the same *y*,

then it is not one-to-one.

(*y* is not unique.)

The horizontal line test is a way

to see if a graph satisfies that definition.

To do the test:

Draw a horizontal line on each point of the graph.

(one unique *y*)

See if there's only one *x* value for each point.

(one *x*)

If so, it is one-to-one.

If not, it's not one-to-one.

It's the horizontal version of the vertical line test.

## Example 1

Draw a horizontal line on each point. (one unique *y*)

Each point shows one *x*.

Then this graph is one-to-one.

## Example 2

(0, 0): one unique *y*, one *x*.

Other points: one ~~unique~~ *y*, two *x*-s.

So this graph is not one-to-one.

## Example 3

If you draw a horizontal line like this,

(one ~~unique~~ *y*)

there are two *x*-s.

So this function is not one-to-one.

## Example 4

Draw *y* = *x*^{2} (*x* ≥ 0)

Quadratic function: vertex form

For every *y* (horizontal line),

there's one *x* (red).

So this graph is one-to-one.