5 Line tests

A line test is a visual test for functions that has the form of $L \subseteq \mathbb{R} \times \mathbb{R}$ . The horizontal axis represent values from the domain, while the vertical axis represent values from the range. A dot (infinitely small) in the 2-D area is an element in $L$ .

Here are the tests and how to interpret the results:

• if you cannot draw a vertical line that intersects with more than one points, $L$ is a function.
• if you can draw a vertical line that intersects with more than one points, $L$ is not a function. In this case, none of the other interpretation is useful.
• if you cannot draw a horizontal line that intersects with more than one points, $L$ is an injection.
• if you can draw a horizontal line that intersects with more than one points, $L$ is not an injection.
• if you cannot draw a horizontal line that does not intersect with any points, $L$ is a surjection.
• if you can draw a horizontal line that does not intersect with any points, $L$ is not a surjection.

Note that line tests are most helpful for continuous mathematical functions like sin, cosine, and etc. They are not particularly helpful in the context of discrete mathematics.