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:
Since every possible vertical line cuts the graph at only one point, $f$ is a function.
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:
Since a vertical line such as $x = 1$ cuts the graph twice, $g$ is not a function.
More on Functions:
Functions
Inverse function
Composite functions
Other concepts (on Functions)