Horizontal Line Test

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: one y (horizontal line), one x. Not one-to-one: one y, not one x.

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

Determine whether the given function is one-to-one.

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

Each point shows one x.

Then this graph is one-to-one.

Example 2

Determine whether the given function is one-to-one.

(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

Determine whether the given function is one-to-one.

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

Determine whether the given function is one-to-one. y = x^2 (x >= 0)

Draw y = x2 (x ≥ 0)

Quadratic function: vertex form

For every y (horizontal line),
there's one x (red).

So this graph is one-to-one.