Final answer:
To determine if a relation is a function without graphing, for a set of ordered pairs, ensure each x-value is associated with only one y-value. For equations, analyze the algebraic form; linear equations represent functions, but equations like those of a circle do not because they associate some x-values with multiple y-values.
Step-by-step explanation:
To determine if a relation is a function without the use of a graph, there are several methods you can use that involve analyzing the set of ordered pairs or the equation itself.
Ordered Pairs
If you have a set of ordered pairs, check each pair to ensure that every x-value (input) corresponds to exactly one y-value (output). If an x-value is paired with more than one y-value, then the relation is not a function.
Equation Analysis
If you have the equation of the relation, analyze its form. For most basic functions, such as linear, quadratic, or polynomial functions, you can predict their graph's behavior based on their algebraic form. A linear equation in the form y = mx + b will always represent a function because for every x-value, there is one unique y-value. However, not all equations that can be graphed as smooth curves are functions. For instance, an equation like x² + y² = r² (which represents a circle) is not a function because most x-values correspond to two y-values (one positive and one negative).
Ultimately, if a relation has precisely one output for every input, it satisfies the definition of a function.