H2 Maths Formulas, Techniques & Graphs >> Functions and Graphs >> Functions >>

Test for Functions by Vertical Line Test

A relation is a function if every input only has one output.

To check if f is a function, verify that any vertical line cuts the graph of f only once.

Example of a function

$$ f : x \mapsto x + 1, x \in \mathbb{R} $$

Graphing the function:

Function f (graph).png

Since every possible vertical line cuts the graph at only one point, $f$ is a function.

Each input to f produces only one output

Each input to f produces only one output

 

Example of a relation (not a function)

$$ g : x \mapsto \pm \sqrt{x + 3}, \phantom{000} x \in \mathbb{R}, x \ge -3 $$

Graphing the relation:

S

Since a vertical line such as $x = 1$ cuts the graph twice, $g$ is not a function.

For example, if 1 is the input to g, 2 outputs are produced

For example, if 1 is the input to g, 2 outputs are produced