Final answer:
A function is a relationship in mathematics where each input is assigned to exactly one output, and it can be proved by using the Vertical Line Test or an Algebraic Method.
Step-by-step explanation:
A function in mathematics describes a specific relationship where each input (often represented as x) is assigned to exactly one output (often represented as y). There are several ways to prove that something is a function:
- Vertical Line Test: Graph the relationship on a coordinate plane and draw vertical lines across the graph. If any vertical line crosses the graph at more than one point, the relationship is not a function because a single input is associated with multiple outputs.
- Algebraic Method: If the relationship is given by an equation, solve for y. If you can express y as a function of x without any ambiguity (i.e., each x-value corresponds to exactly one y-value), then it is a function.
Mathematical functions are critical in fields like economics where they often describe cause and effect relationships or definitions as part of economic models. They are used to explain how one variable influences another, such as how your GPA might depend on the grades received in courses.