Horizontal Line Test

Horizontal Line Test

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

One-to-one

One-to-one means [one input], [one unique output]. So, if each x is mapped to a unique y, then the function is one-to-one.

One-to-one means [one input], [one unique output].

So, if each x is mapped to a unique y,
then the function is one-to-one.

Not One-to-one

If two or more x-s are mapped to the same y, then the function is not one-to-one.

If two or more x-s are mapped to the same y,
then the function is not one-to-one.

In this picture,
x1 and x2 are mapped to the same y1.
So this function is not one-to-one.

How to Test

If every horizontal line shows one input, then the function passes the test. Then the function is one-to-one.

To do the test:

Make a horizontal line
on each point of the graph.

The horizontal line means [one unique output], y.
(To make a horizontal line,
you can use your pencil, ruler, etc.)

Then see if each vertical line shows [one input], x.

If every horizontal line shows one input,
then the graph passes the test.

Then the function is one-to-one.

Just think it as
the horizontal version of the vertical line test.

Vertical line test

Actually, the horizontal line test is also used
to see if a function has an inverse function.

To find out the details,
see the inverse function page below.

Inverse functions - Relationship between the horizontal line test

If there's any horizontal line that does not show one input, then the function fails the test. Then the function is not one-to-one.

If there's any horizontal line
that does not show [one input],
then the function fails the test.

Then the function is not one-to-one.

Example 1

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

Starting from the bottom,
do the horizontal line test.

Every horizontal line shows one input.

So the function passes the horizontal line test.

Then the function is one-to-one.

Example 2

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

At the origin,
the horizontal line shows one input.

So, at the origin,
the function passes the horizontal line test.

But if you make a horizontal line like this,
it shows two inputs.

So the graph fails the horizontal line test.

So this function is not one-to-one.

Example 3

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

If you make a horizontal line like this,
the horizontal line shows one input.

So, for this y value,
the function passes the horizontal line test.

But if you make a horizontal line like this,
it shows two inputs.

So the graph fails the horizontal line test.

So this function is not one-to-one.