Horizontal Line Test

How to do the horizontal line test to see if a graph is a one-to-one function: 2 examples and their solutions.

One-to-one

One-to-one

A one-to-one function means
one x, one unique y.

So, if x1 is mapped to y1,
then the other x values are not mapped to y1.

Not One-to-one

See this map.
Both x1 and x2 are mapped to the same y1.

Then this is not one-to-one.

(Both x1 and x2 satisfy
[one x, one y].
So this is a function.)

Horizontal Line Test

Pass

A horizontal line test is a way
to see if a graph is a one-to-one function.
(to see if a graph shows [one x, one unique y])

For every point of the graph,
draw a horizontal line (= one unique y).

If the horizontal line and the graph
intersect at one point (= one x),
the graph passes the test.

Fail

If the horizontal line and the graph
intersect at more than one point
(= not one x = not one unique y),

the graph fails the test.

Example 1

Example

Solution

To see if the graph is a one-to-one function,
first, see if the graph is a function
by doing the vertical line test.

If the graph fails the vertical line test,
then it cannot be a function.

From the left side of the graph,
think of a vertical line.
Move the vertical line to the right side.
And see if the vertical line and the graph
intersect at [one point].

As you can see,
the line and the graph intersect at one point.

So this graph passes the vertical line test.

So the graph is a function.

Next, to see if the function is one-to-one,
do the horizontal line test.

From the bottom side of the graph,
think of a horizontal line.
Move the horizontal line to the top side.
And see if the horizontal line and the graph
intersect at [one point].

As you can see,
the line and the graph intersect at one point.

So this graph passes the horizontal line test.

So the function is one-to-one.

The graph is a function.
And it is one-to-one.

So the graph is a one-to-one function.

So
One-to-one function
is the answer.

Example 2

Example

Solution

First, to see if the graph is a function,
do the vertical line test.

From the left side of the graph,
think of a vertical line.
Move the vertical line to the right side.
And see if the vertical line and the graph
intersect at [one point].

As you can see,
the line and the graph intersect at one point.

So this graph passes the vertical line test.

So the graph is a function.

Next, to see if the function is one-to-one,
do the horizontal line test.

If the horizontal line is like this,
the line and the graph intersect at one point.

So, for this case,
the graph passes the test.

But if the horizontal line is like this,
the line and the graph intersect at two points:
not at one point.

So this graph fails the horizontal line test.

So the function is not one-to-one.

The graph is a function.
It is not one-to-one.

So the graph is not a one-to-one function.

So
Not a one-to-one function
is the answer.